:: Propositional Calculus for Boolean Valued Functions, II :: by Shunichi Kobayashi and Yatsuka Nakamura :: :: Received March 13, 1999 :: Copyright (c) 1999-2017 Association of Mizar Users :: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland). :: This code can be distributed under the GNU General Public Licence :: version 3.0 or later, or the Creative Commons Attribution-ShareAlike :: License version 3.0 or later, subject to the binding interpretation :: detailed in file COPYING.interpretation. :: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these :: licenses, or see http://www.gnu.org/licenses/gpl.html and :: http://creativecommons.org/licenses/by-sa/3.0/. environ vocabularies XBOOLE_0, SUBSET_1, MARGREL1, BVFUNC_1, XBOOLEAN, FUNCT_1, PARTIT1; notations XBOOLE_0, SUBSET_1, XBOOLEAN, MARGREL1, FUNCT_2, BVFUNC_1; constructors BINARITH, BVFUNC_1, XBOOLEAN, NUMBERS; registrations XBOOLEAN, MARGREL1, ORDINAL1; requirements ARITHM, BOOLE, NUMERALS; definitions BVFUNC_1, FUNCT_2; equalities XBOOLEAN, MARGREL1; expansions XBOOLEAN, BVFUNC_1; theorems MARGREL1, BINARITH, BVFUNC_1, XBOOLEAN, BVFUNC_4, BVFUNC_5; begin reserve Y for non empty set; theorem for a,b being Function of Y,BOOLEAN holds a 'imp' (b 'imp' (a '&' b))=I_el(Y) proof let a,b be Function of Y,BOOLEAN; for x being Element of Y holds (a 'imp' (b 'imp' (a '&' b))).x=TRUE proof let x be Element of Y; (a 'imp' (b 'imp' (a '&' b))).x ='not' a.x 'or' (b 'imp' (a '&' b)). x by BVFUNC_1:def 8 .='not' a.x 'or' ('not' b.x 'or' (a '&' b).x) by BVFUNC_1:def 8 .='not' a.x 'or' ('not' b.x 'or' (a.x '&' b.x)) by MARGREL1:def 20 .='not' a.x 'or' (('not' b.x 'or' a.x) '&' ('not' b.x 'or' (b) .x)) by XBOOLEAN:9 .='not' a.x 'or' (TRUE '&' ('not' b.x 'or' a.x)) by XBOOLEAN:102 .=('not' a.x 'or' a.x) 'or' 'not' b.x .=TRUE 'or' 'not' b.x by XBOOLEAN:102 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b being Function of Y,BOOLEAN holds (a 'imp' b) 'imp' ((b 'imp' a) 'imp' (a 'eqv' b))=I_el(Y) proof let a,b be Function of Y,BOOLEAN; for x being Element of Y holds ((a 'imp' b) 'imp' ((b 'imp' a) 'imp' (a 'eqv' b))).x=TRUE proof let x be Element of Y; ((a 'imp' b) 'imp' ((b 'imp' a) 'imp' (a 'eqv' b))).x ='not' (a 'imp' b).x 'or' ((b 'imp' a) 'imp' (a 'eqv' b)).x by BVFUNC_1:def 8 .='not'( 'not' a.x 'or' b.x) 'or' ((b 'imp' a) 'imp' (a 'eqv' b)). x by BVFUNC_1:def 8 .='not'( 'not' a.x 'or' b.x) 'or' ('not' (b 'imp' a).x 'or' (a 'eqv' b).x) by BVFUNC_1:def 8 .=('not' 'not' a.x '&' 'not' b.x) 'or' ('not'( 'not' b.x 'or' (a ).x) 'or' (a 'eqv' b).x) by BVFUNC_1:def 8 .=(a.x '&' 'not' b.x) 'or' ((b.x '&' 'not' a.x) 'or' 'not'( (a ).x 'xor' b.x)) by BVFUNC_1:def 9 .=(a.x '&' 'not' b.x) 'or' ((('not' a.x '&' b.x) 'or' 'not'( 'not' a.x '&' b.x)) '&' (('not' a.x '&' b.x) 'or' 'not'( a.x '&' 'not' b.x))) by XBOOLEAN:9 .=(a.x '&' 'not' b.x) 'or' (TRUE '&' (('not' a.x '&' b.x) 'or' 'not'( a.x '&' 'not' b.x))) by XBOOLEAN:102 .=((a.x '&' 'not' b.x) 'or' 'not'( a.x '&' 'not' b.x)) 'or' ( 'not' a.x '&' b.x) .=TRUE 'or' ('not' a.x '&' b.x) by XBOOLEAN:102 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b being Function of Y,BOOLEAN holds (a 'or' b) 'eqv' (b 'or' a)=I_el(Y) proof let a,b be Function of Y,BOOLEAN; for x being Element of Y holds ((a 'or' b) 'eqv' (b 'or' a)).x=TRUE proof let x be Element of Y; ((a 'or' b) 'eqv' (b 'or' a)).x ='not'( (a 'or' b).x 'xor' (b 'or' a). x) by BVFUNC_1:def 9 .=TRUE by XBOOLEAN:102; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b,c being Function of Y,BOOLEAN holds ((a '&' b) 'imp' c) 'imp' (a 'imp' (b 'imp' c))=I_el(Y) proof let a,b,c be Function of Y,BOOLEAN; for x being Element of Y holds (((a '&' b) 'imp' c) 'imp' (a 'imp' (b 'imp' c))).x=TRUE proof let x be Element of Y; (((a '&' b) 'imp' c) 'imp' (a 'imp' (b 'imp' c))).x ='not' ((a '&' b) 'imp' c).x 'or' (a 'imp' (b 'imp' c)).x by BVFUNC_1:def 8 .='not'( 'not' (a '&' b).x 'or' c.x) 'or' (a 'imp' (b 'imp' c)).x by BVFUNC_1:def 8 .='not'( 'not' (a.x '&' b.x) 'or' c.x) 'or' (a 'imp' (b 'imp' c) ).x by MARGREL1:def 20 .='not'( 'not'( a.x '&' b.x) 'or' c.x) 'or' ('not' a.x 'or' (b 'imp' c).x) by BVFUNC_1:def 8 .='not'( ('not' a.x 'or' 'not' b.x) 'or' c.x) 'or' ('not' a.x 'or' ('not' b.x 'or' c.x)) by BVFUNC_1:def 8 .=TRUE by XBOOLEAN:102; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'imp' (b 'imp' c) ) 'imp' ((a '&' b) 'imp' c)=I_el(Y) proof let a,b,c be Function of Y,BOOLEAN; for x being Element of Y holds ((a 'imp' (b 'imp' c)) 'imp' ((a '&' b) 'imp' c)).x=TRUE proof let x be Element of Y; ((a 'imp' (b 'imp' c)) 'imp' ((a '&' b) 'imp' c)).x ='not' (a 'imp' (b 'imp' c)).x 'or' ((a '&' b) 'imp' c).x by BVFUNC_1:def 8 .='not'( 'not' a.x 'or' (b 'imp' c).x) 'or' ((a '&' b) 'imp' c).x by BVFUNC_1:def 8 .='not'( 'not' a.x 'or' ('not' b.x 'or' c.x)) 'or' ((a '&' b) 'imp' c).x by BVFUNC_1:def 8 .='not'( 'not' a.x 'or' ('not' b.x 'or' c.x)) 'or' ('not' (a '&' b).x 'or' c.x) by BVFUNC_1:def 8 .='not'( 'not' a.x 'or' ('not' b.x 'or' c.x)) 'or' (('not' a.x 'or' 'not' b.x) 'or' c.x) by MARGREL1:def 20 .=TRUE by XBOOLEAN:102; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b,c being Function of Y,BOOLEAN holds (c 'imp' a) 'imp' (( c 'imp' b) 'imp' (c 'imp' (a '&' b)))=I_el(Y) proof let a,b,c be Function of Y,BOOLEAN; for x being Element of Y holds ((c 'imp' a) 'imp' ((c 'imp' b) 'imp' (c 'imp' (a '&' b)))).x=TRUE proof let x be Element of Y; ((c 'imp' a) 'imp' ((c 'imp' b) 'imp' (c 'imp' (a '&' b)))).x ='not' ( c 'imp' a).x 'or' ((c 'imp' b) 'imp' (c 'imp' (a '&' b))).x by BVFUNC_1:def 8 .='not'( 'not' c.x 'or' a.x) 'or' ((c 'imp' b) 'imp' (c 'imp' (a '&' b))).x by BVFUNC_1:def 8 .='not'( 'not' c.x 'or' a.x) 'or' ('not' (c 'imp' b).x 'or' (c 'imp' (a '&' b)).x) by BVFUNC_1:def 8 .='not'( 'not' c.x 'or' a.x) 'or' ('not'( 'not' c.x 'or' b.x) 'or' (c 'imp' (a '&' b)).x) by BVFUNC_1:def 8 .='not'( 'not' c.x 'or' a.x) 'or' ('not'( 'not' c.x 'or' b.x) 'or' ('not' c.x 'or' (a '&' b).x)) by BVFUNC_1:def 8 .=(c.x '&' 'not' a.x) 'or' (('not' 'not' c.x '&' 'not' b.x) 'or' ('not' c.x 'or' (a.x '&' b.x))) by MARGREL1:def 20 .=(c.x '&' 'not' a.x) 'or' ((c.x '&' 'not' b.x) 'or' (('not' ( c).x 'or' a.x) '&' ('not' c.x 'or' b.x))) by XBOOLEAN:9 .=(c.x '&' 'not' a.x) 'or' (((c.x '&' 'not' b.x) 'or' ('not' ( c).x 'or' a.x)) '&' ((c.x '&' 'not' b.x) 'or' ('not' c.x 'or' 'not' 'not' b.x))) by XBOOLEAN:9 .=(c.x '&' 'not' a.x) 'or' (TRUE '&' ((c.x '&' 'not' b.x) 'or' ('not' c.x 'or' a.x))) by XBOOLEAN:102 .=((c.x '&' 'not' a.x) 'or' 'not'( c.x '&' 'not' a.x)) 'or' (( c).x '&' 'not' b.x) .=TRUE 'or' (c.x '&' 'not' b.x) by XBOOLEAN:102 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b,c being Function of Y,BOOLEAN holds ((a 'or' b) 'imp' c) 'imp' ((a 'imp' c) 'or' (b 'imp' c))=I_el(Y) proof let a,b,c be Function of Y,BOOLEAN; for x being Element of Y holds (((a 'or' b) 'imp' c) 'imp' ((a 'imp' c) 'or' (b 'imp' c))).x=TRUE proof let x be Element of Y; (((a 'or' b) 'imp' c) 'imp' ((a 'imp' c) 'or' (b 'imp' c))).x ='not' ( (a 'or' b) 'imp' c).x 'or' ((a 'imp' c) 'or' (b 'imp' c)).x by BVFUNC_1:def 8 .='not'( 'not' (a 'or' b).x 'or' c.x) 'or' ((a 'imp' c) 'or' (b 'imp' c)).x by BVFUNC_1:def 8 .='not'( 'not'( a.x 'or' b.x) 'or' c.x) 'or' ((a 'imp' c) 'or' ( b 'imp' c)).x by BVFUNC_1:def 4 .='not'( 'not'( a.x 'or' b.x) 'or' c.x) 'or' ((a 'imp' c).x 'or' (b 'imp' c).x) by BVFUNC_1:def 4 .='not'( 'not'( a.x 'or' b.x) 'or' c.x) 'or' (('not' a.x 'or' c.x) 'or' (b 'imp' c).x) by BVFUNC_1:def 8 .=('not' 'not'( a.x 'or' b.x) '&' 'not' c.x) 'or' (('not' a.x 'or' c.x) 'or' ('not' b.x 'or' c.x)) by BVFUNC_1:def 8 .=((b.x '&' 'not' c.x) 'or' (a.x '&' 'not' c.x)) 'or' ('not'( a.x '&' 'not' c.x) 'or' ('not' b.x 'or' c.x)) by XBOOLEAN:8 .=((b.x '&' 'not' c.x) 'or' ((a.x '&' 'not' c.x) 'or' 'not'( ( a).x '&' 'not' c.x))) 'or' ('not' b.x 'or' c.x) .=((b.x '&' 'not' c.x) 'or' TRUE) 'or' ('not' b.x 'or' c.x) by XBOOLEAN:102 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'imp' c) 'imp' (( b 'imp' c) 'imp' ((a 'or' b) 'imp' c))=I_el(Y) proof let a,b,c be Function of Y,BOOLEAN; for x being Element of Y holds ((a 'imp' c) 'imp' ((b 'imp' c) 'imp' ((a 'or' b) 'imp' c))).x=TRUE proof let x be Element of Y; ((a 'imp' c) 'imp' ((b 'imp' c) 'imp' ((a 'or' b) 'imp' c))).x ='not' (a 'imp' c).x 'or' ((b 'imp' c) 'imp' ((a 'or' b) 'imp' c)).x by BVFUNC_1:def 8 .='not'( 'not' a.x 'or' c.x) 'or' ((b 'imp' c) 'imp' ((a 'or' b) 'imp' c)).x by BVFUNC_1:def 8 .='not'( 'not' a.x 'or' c.x) 'or' ('not' (b 'imp' c).x 'or' ((a 'or' b) 'imp' c).x) by BVFUNC_1:def 8 .='not'( 'not' a.x 'or' c.x) 'or' ('not'( 'not' b.x 'or' c.x) 'or' ((a 'or' b) 'imp' c).x) by BVFUNC_1:def 8 .='not'( 'not' a.x 'or' c.x) 'or' ('not'( 'not' b.x 'or' c.x) 'or' ('not' (a 'or' b).x 'or' c.x)) by BVFUNC_1:def 8 .='not'( 'not' a.x 'or' c.x) 'or' ('not'( 'not' b.x 'or' c.x) 'or' ('not' (a.x 'or' b.x) 'or' c.x)) by BVFUNC_1:def 4 .='not'( 'not' a.x 'or' c.x) 'or' ('not'( 'not' b.x 'or' c.x) 'or' ((c.x 'or' 'not' a.x) '&' ('not' b.x 'or' c.x))) by XBOOLEAN:9 .='not'( 'not' a.x 'or' c.x) 'or' (('not'( 'not' b.x 'or' c.x) 'or' (c.x 'or' 'not' a.x)) '&' ('not'( 'not' b.x 'or' c.x) 'or' ('not' b.x 'or' c.x))) by XBOOLEAN:9 .='not'( 'not' a.x 'or' c.x) 'or' (TRUE '&' ('not'( 'not' b.x 'or' c.x) 'or' (c.x 'or' 'not' a.x))) by XBOOLEAN:102 .='not'( 'not' a.x 'or' c.x) 'or' ('not'( 'not' b.x 'or' c.x) 'or' ('not' a.x 'or' c.x)) .=('not'( 'not' a.x 'or' c.x) 'or' ('not' a.x 'or' c.x)) 'or' 'not'( 'not' b.x 'or' c.x) .=TRUE 'or' 'not'( 'not' b.x 'or' c.x) by XBOOLEAN:102 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem Th9: for a,b,c being Function of Y,BOOLEAN holds ((a 'imp' c) '&' (b 'imp' c)) 'imp' ((a 'or' b) 'imp' c)=I_el(Y) proof let a,b,c be Function of Y,BOOLEAN; for x being Element of Y holds (((a 'imp' c) '&' (b 'imp' c)) 'imp' ((a 'or' b) 'imp' c)).x=TRUE proof let x be Element of Y; (((a 'imp' c) '&' (b 'imp' c)) 'imp' ((a 'or' b) 'imp' c)).x ='not' (( a 'imp' c) '&' (b 'imp' c)).x 'or' ((a 'or' b) 'imp' c).x by BVFUNC_1:def 8 .='not'( (a 'imp' c).x '&' (b 'imp' c).x) 'or' ((a 'or' b) 'imp' c).x by MARGREL1:def 20 .='not'( ('not' a.x 'or' c.x) '&' (b 'imp' c).x) 'or' ((a 'or' b) 'imp' c).x by BVFUNC_1:def 8 .='not'( ('not' a.x 'or' c.x) '&' ('not' b.x 'or' c.x)) 'or' ( (a 'or' b) 'imp' c).x by BVFUNC_1:def 8 .='not'( ('not' a.x 'or' c.x) '&' ('not' b.x 'or' c.x)) 'or' ( 'not' (a 'or' b).x 'or' c.x) by BVFUNC_1:def 8 .=('not'( 'not' a.x 'or' c.x) 'or' 'not'( 'not' b.x 'or' c.x)) 'or' (c.x 'or' 'not'( a.x 'or' b.x)) by BVFUNC_1:def 4 .=('not'( 'not' a.x 'or' c.x) 'or' 'not'( 'not' b.x 'or' c.x)) 'or' (('not' a.x 'or' c.x) '&' (c.x 'or' 'not' b.x)) by XBOOLEAN:9 .=(('not'( 'not' a.x 'or' c.x) 'or' 'not'( 'not' b.x 'or' c.x) ) 'or' ('not' a.x 'or' c.x)) '&' (('not'( 'not' a.x 'or' c.x) 'or' 'not'( 'not' b.x 'or' c.x)) 'or' ('not' b.x 'or' c.x)) by XBOOLEAN:9 .=('not'( 'not' b.x 'or' c.x) 'or' ('not'( 'not' a.x 'or' c.x) 'or' ('not' a.x 'or' c.x))) '&' ('not'( 'not' a.x 'or' c.x) 'or' ('not' ( 'not' b.x 'or' c.x) 'or' ('not' b.x 'or' c.x))) .=('not'( 'not' b.x 'or' c.x) 'or' TRUE) '&' ('not'( 'not' a.x 'or' c.x) 'or' ('not'( 'not' b.x 'or' c.x) 'or' ('not' b.x 'or' c.x)) ) by XBOOLEAN:102 .=TRUE '&' ('not'( 'not' a.x 'or' c.x) 'or' TRUE) by XBOOLEAN:102 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b being Function of Y,BOOLEAN holds (a 'imp' (b '&' 'not' b)) 'imp' 'not' a=I_el(Y) proof let a,b be Function of Y,BOOLEAN; for x being Element of Y holds ((a 'imp' (b '&' 'not' b)) 'imp' 'not' a) .x=TRUE proof let x be Element of Y; ((a 'imp' (b '&' 'not' b)) 'imp' 'not' a).x ='not' (a 'imp' (b '&' 'not' b)).x 'or' ('not' a).x by BVFUNC_1:def 8 .='not'( 'not' a.x 'or' (b '&' 'not' b).x) 'or' ('not' a).x by BVFUNC_1:def 8 .=(a.x '&' ('not' b.x 'or' 'not' ('not' b).x)) 'or' ('not' a).x by MARGREL1:def 20 .=(a.x '&' ('not' b.x 'or' 'not' 'not' b.x)) 'or' ('not' a).x by MARGREL1:def 19 .=(a.x '&' TRUE) 'or' ('not' a).x by XBOOLEAN:102 .=a.x 'or' 'not' a.x by MARGREL1:def 19 .=TRUE by XBOOLEAN:102; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b,c being Function of Y,BOOLEAN holds ((a 'or' b) '&' (a 'or' c)) 'imp' (a 'or' (b '&' c))=I_el(Y) proof let a,b,c be Function of Y,BOOLEAN; for x being Element of Y holds (((a 'or' b) '&' (a 'or' c)) 'imp' (a 'or' (b '&' c))).x=TRUE proof let x be Element of Y; (((a 'or' b) '&' (a 'or' c)) 'imp' (a 'or' (b '&' c))).x ='not' ((a 'or' b) '&' (a 'or' c)).x 'or' (a 'or' (b '&' c)).x by BVFUNC_1:def 8 .='not'( (a 'or' b).x '&' (a 'or' c).x) 'or' (a 'or' (b '&' c)).x by MARGREL1:def 20 .='not' ((a.x 'or' b.x) '&' (a 'or' c).x) 'or' (a 'or' (b '&' c)). x by BVFUNC_1:def 4 .='not'( (a.x 'or' b.x) '&' (a.x 'or' c.x)) 'or' (a 'or' (b '&' c)).x by BVFUNC_1:def 4 .='not'( (a.x 'or' b.x) '&' (a.x 'or' c.x)) 'or' (a.x 'or' ( b '&' c).x) by BVFUNC_1:def 4 .='not'( (a.x 'or' b.x) '&' (a.x 'or' c.x)) 'or' (a.x 'or' ( b.x '&' c.x)) by MARGREL1:def 20 .=('not'( a.x 'or' b.x) 'or' 'not'( a.x 'or' c.x)) 'or' ((a. x 'or' b.x) '&' (a.x 'or' c.x)) by XBOOLEAN:9 .=(('not'( a.x 'or' c.x) 'or' 'not'( a.x 'or' b.x)) 'or' (a. x 'or' b.x)) '&' (('not'( a.x 'or' b.x) 'or' 'not'( a.x 'or' c.x)) 'or' (a.x 'or' c.x)) by XBOOLEAN:9 .=('not'( a.x 'or' c.x) 'or' ('not'( a.x 'or' b.x) 'or' (a.x 'or' b.x))) '&' ('not'( a.x 'or' b.x) 'or' ('not'( a.x 'or' c.x) 'or' (a.x 'or' c.x))) .=('not'( a.x 'or' c.x) 'or' TRUE) '&' ('not'( a.x 'or' b.x) 'or' ('not'( a.x 'or' c.x) 'or' (a.x 'or' c.x))) by XBOOLEAN:102 .=TRUE '&' ('not'( a.x 'or' b.x) 'or' TRUE) by XBOOLEAN:102 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a '&' (b 'or' c)) 'imp' ((a '&' b) 'or' (a '&' c))=I_el(Y) proof let a,b,c be Function of Y,BOOLEAN; for x being Element of Y holds ((a '&' (b 'or' c)) 'imp' ((a '&' b) 'or' (a '&' c))).x=TRUE proof let x be Element of Y; ((a '&' (b 'or' c)) 'imp' ((a '&' b) 'or' (a '&' c))).x ='not' (a '&' (b 'or' c)).x 'or' ((a '&' b) 'or' (a '&' c)).x by BVFUNC_1:def 8 .='not'( a.x '&' (b 'or' c).x) 'or' ((a '&' b) 'or' (a '&' c)).x by MARGREL1:def 20 .='not' (a.x '&' (b.x 'or' c.x)) 'or' ((a '&' b) 'or' (a '&' c)) .x by BVFUNC_1:def 4 .='not'( a.x '&' (b.x 'or' c.x)) 'or' ((a '&' b).x 'or' (a '&' c ).x) by BVFUNC_1:def 4 .='not'( a.x '&' (b.x 'or' c.x)) 'or' ((a.x '&' b.x) 'or' (a '&' c).x) by MARGREL1:def 20 .='not'( a.x '&' (b.x 'or' c.x)) 'or' ((a.x '&' b.x) 'or' (( a).x '&' c.x)) by MARGREL1:def 20 .= ((a.x '&' b.x) 'or' (a.x '&' c.x)) 'or' ('not'( a.x '&' ( b).x) '&' 'not' (a.x '&' c.x)) by XBOOLEAN:8 .= (((a.x '&' c.x) 'or' (a.x '&' b.x)) 'or' 'not'( a.x '&' ( b).x)) '&' (((a.x '&' b.x) 'or' (a.x '&' c.x)) 'or' 'not'( a.x '&' (c ).x)) by XBOOLEAN:9 .= ((a.x '&' c.x) 'or' ((a.x '&' b.x) 'or' 'not'( a.x '&' (b ).x))) '&' ((a.x '&' b.x) 'or' ((a.x '&' c.x) 'or' 'not'( a.x '&' (c) .x))) .= ((a.x '&' c.x) 'or' TRUE) '&' ((a.x '&' b.x) 'or' ((a.x '&' c.x) 'or' 'not'( a.x '&' c.x))) by XBOOLEAN:102 .= TRUE '&' ((a.x '&' b.x) 'or' TRUE) by XBOOLEAN:102 .= TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b,c being Function of Y,BOOLEAN holds ((a 'or' c) '&' (b 'or' c)) 'imp' ((a '&' b) 'or' c)=I_el(Y) proof let a,b,c be Function of Y,BOOLEAN; for x being Element of Y holds (((a 'or' c) '&' (b 'or' c)) 'imp' ((a '&' b) 'or' c)).x=TRUE proof let x be Element of Y; (((a 'or' c) '&' (b 'or' c)) 'imp' ((a '&' b) 'or' c)).x ='not' ((a 'or' c) '&' (b 'or' c)).x 'or' ((a '&' b) 'or' c).x by BVFUNC_1:def 8 .='not'( (a 'or' c).x '&' (b 'or' c).x) 'or' ((a '&' b) 'or' c).x by MARGREL1:def 20 .='not' ((a.x 'or' c.x) '&' (b 'or' c).x) 'or' ((a '&' b) 'or' c). x by BVFUNC_1:def 4 .=('not'( a.x 'or' c.x) 'or' 'not'( b.x 'or' c.x)) 'or' ((a '&' b) 'or' c).x by BVFUNC_1:def 4 .=('not'( a.x 'or' c.x) 'or' 'not'( b.x 'or' c.x)) 'or' ((a '&' b).x 'or' c.x) by BVFUNC_1:def 4 .=('not'( a.x 'or' c.x) 'or' 'not'( b.x 'or' c.x)) 'or' (c.x 'or' (a.x '&' b.x)) by MARGREL1:def 20 .=('not'( a.x 'or' c.x) 'or' 'not'( b.x 'or' c.x)) 'or' ((a. x 'or' c.x) '&' (c.x 'or' b.x)) by XBOOLEAN:9 .=(('not'( a.x 'or' c.x) 'or' 'not'( b.x 'or' c.x)) 'or' (a. x 'or' c.x)) '&' (('not'( a.x 'or' c.x) 'or' 'not'( b.x 'or' c.x)) 'or' (b.x 'or' c.x)) by XBOOLEAN:9 .=('not'( b.x 'or' c.x) 'or' ('not'( a.x 'or' c.x) 'or' (a.x 'or' c.x))) '&' ('not'( a.x 'or' c.x) 'or' ('not'( b.x 'or' c.x) 'or' (b.x 'or' c.x))) .=('not'( b.x 'or' c.x) 'or' TRUE) '&' ('not'( a.x 'or' c.x) 'or' ('not'( b.x 'or' c.x) 'or' (b.x 'or' c.x))) by XBOOLEAN:102 .=TRUE '&' ('not'( a.x 'or' c.x) 'or' TRUE) by XBOOLEAN:102 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b,c being Function of Y,BOOLEAN holds ((a 'or' b) '&' c) 'imp' ((a '&' c) 'or' (b '&' c))=I_el(Y) proof let a,b,c be Function of Y,BOOLEAN; for x being Element of Y holds (((a 'or' b) '&' c) 'imp' ((a '&' c) 'or' (b '&' c))).x=TRUE proof let x be Element of Y; (((a 'or' b) '&' c) 'imp' ((a '&' c) 'or' (b '&' c))).x ='not' ((a 'or' b) '&' c).x 'or' ((a '&' c) 'or' (b '&' c)).x by BVFUNC_1:def 8 .='not'( ((a 'or' b)).x '&' c.x) 'or' ((a '&' c) 'or' (b '&' c)).x by MARGREL1:def 20 .='not'( (a.x 'or' b.x) '&' c.x) 'or' ((a '&' c) 'or' (b '&' c)) .x by BVFUNC_1:def 4 .='not'( (a.x 'or' b.x) '&' c.x) 'or' ((a '&' c).x 'or' (b '&' c ).x) by BVFUNC_1:def 4 .='not'( (a.x 'or' b.x) '&' c.x) 'or' ((a.x '&' c.x) 'or' (b '&' c).x) by MARGREL1:def 20 .='not'( c.x '&' (a.x 'or' b.x)) 'or' ((a.x '&' c.x) 'or' (( b).x '&' c.x)) by MARGREL1:def 20 .=((a.x '&' c.x) 'or' (b.x '&' c.x)) 'or' ('not'( a.x '&' (c ).x) '&' 'not'( b.x '&' c.x)) by XBOOLEAN:8 .=(((b.x '&' c.x) 'or' (a.x '&' c.x)) 'or' 'not'( a.x '&' (c ).x)) '&' (((a.x '&' c.x) 'or' (b.x '&' c.x)) 'or' 'not'( b.x '&' (c) .x)) by XBOOLEAN:9 .=((b.x '&' c.x) 'or' ((a.x '&' c.x) 'or' 'not'( a.x '&' (c) .x))) '&' ((a.x '&' c.x) 'or' ((b.x '&' c.x) 'or' 'not'( b.x '&' c. x))) .=((b.x '&' c.x) 'or' TRUE) '&' ((a.x '&' c.x) 'or' ((b.x '&' c.x) 'or' 'not'( b.x '&' c.x))) by XBOOLEAN:102 .=TRUE '&' ((a.x '&' c.x) 'or' TRUE) by XBOOLEAN:102 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b being Function of Y,BOOLEAN holds (a '&' b)=I_el(Y) implies (a 'or' b)=I_el(Y) proof let a,b be Function of Y,BOOLEAN; assume A1: (a '&' b)=I_el(Y); for x being Element of Y holds (a 'or' b).x=TRUE proof let x be Element of Y; (a '&' b).x= TRUE by A1,BVFUNC_1:def 11; then A2: a.x '&' b.x=TRUE by MARGREL1:def 20; then a.x=TRUE by MARGREL1:12; then (a 'or' b).x =TRUE 'or' TRUE by A2,BVFUNC_1:def 4 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b)=I_el(Y) implies (a 'or' c) 'imp' (b 'or' c)=I_el(Y) proof let a,b,c be Function of Y,BOOLEAN; assume A1: (a 'imp' b)=I_el(Y); for x being Element of Y holds ((a 'or' c) 'imp' (b 'or' c)).x=TRUE proof let x be Element of Y; (a 'imp' b).x= TRUE by A1,BVFUNC_1:def 11; then A2: 'not' a.x 'or' b.x = TRUE by BVFUNC_1:def 8; ((a 'or' c) 'imp' (b 'or' c)).x ='not' (a 'or' c).x 'or' (b 'or' c).x by BVFUNC_1:def 8 .='not'( a.x 'or' c.x) 'or' (b 'or' c).x by BVFUNC_1:def 4 .=(b.x 'or' c.x) 'or' ('not' a.x '&' 'not' c.x) by BVFUNC_1:def 4 .=((c.x 'or' b.x) 'or' 'not' a.x) '&' ((b.x 'or' c.x) 'or' 'not' c.x) by XBOOLEAN:9 .=(c.x 'or' ('not' a.x 'or' b.x)) '&' ((b.x 'or' c.x) 'or' 'not' c.x) .=TRUE '&' (b.x 'or' (c.x 'or' 'not' c.x)) by A2 .=TRUE '&' (b.x 'or' TRUE) by XBOOLEAN:102 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b)=I_el(Y) implies (a '&' c) 'imp' (b '&' c)=I_el(Y) proof let a,b,c be Function of Y,BOOLEAN; assume A1: (a 'imp' b)=I_el(Y); for x being Element of Y holds ((a '&' c) 'imp' (b '&' c)).x=TRUE proof let x be Element of Y; (a 'imp' b).x= TRUE by A1,BVFUNC_1:def 11; then A2: 'not' a.x 'or' b.x = TRUE by BVFUNC_1:def 8; ((a '&' c) 'imp' (b '&' c)).x ='not' (a '&' c).x 'or' (b '&' c).x by BVFUNC_1:def 8 .='not'( a.x '&' c.x) 'or' (b '&' c).x by MARGREL1:def 20 .=('not' a.x 'or' 'not' c.x) 'or' (b.x '&' c.x) by MARGREL1:def 20 .=(('not' c.x 'or' 'not' a.x) 'or' b.x) '&' (('not' a.x 'or' 'not' c.x) 'or' c.x) by XBOOLEAN:9 .=('not' c.x 'or' ('not' a.x 'or' b.x)) '&' ('not' a.x 'or' ( 'not' c.x 'or' c.x)) .=('not' c.x 'or' ('not' a.x 'or' b.x)) '&' ('not' a.x 'or' TRUE) by XBOOLEAN:102 .=TRUE by A2; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem th18: for a,b,c being Function of Y,BOOLEAN holds (c 'imp' a)=I_el(Y) & (c 'imp' b)=I_el(Y) implies c 'imp' (a '&' b)=I_el(Y) proof let a,b,c be Function of Y,BOOLEAN; assume that A1: (c 'imp' a)=I_el(Y) and A2: (c 'imp' b)=I_el(Y); for x being Element of Y holds (c 'imp' (a '&' b)).x=TRUE proof let x be Element of Y; (c 'imp' a).x= TRUE by A1,BVFUNC_1:def 11; then A3: 'not' c.x 'or' a.x = TRUE by BVFUNC_1:def 8; (c 'imp' b).x= TRUE by A2,BVFUNC_1:def 11; then A4: 'not' c.x 'or' b.x = TRUE by BVFUNC_1:def 8; (c 'imp' (a '&' b)).x ='not' c.x 'or' (a '&' b).x by BVFUNC_1:def 8 .='not' c.x 'or' (a.x '&' b.x) by MARGREL1:def 20 .=TRUE '&' TRUE by A3,A4,XBOOLEAN:9 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'imp' c)=I_el(Y) & (b 'imp' c)=I_el(Y) implies (a 'or' b) 'imp' c = I_el(Y) proof let a,b,c be Function of Y,BOOLEAN; assume that A1: (a 'imp' c)=I_el(Y) and A2: (b 'imp' c)=I_el(Y); for x being Element of Y holds ((a 'or' b) 'imp' c).x=TRUE proof let x be Element of Y; (a 'imp' c).x= TRUE by A1,BVFUNC_1:def 11; then A3: 'not' a.x 'or' c.x = TRUE by BVFUNC_1:def 8; (b 'imp' c).x= TRUE by A2,BVFUNC_1:def 11; then A4: 'not' b.x 'or' c.x = TRUE by BVFUNC_1:def 8; ((a 'or' b) 'imp' c).x ='not' (a 'or' b).x 'or' c.x by BVFUNC_1:def 8 .='not'( a.x 'or' b.x) 'or' c.x by BVFUNC_1:def 4 .=TRUE '&' TRUE by A3,A4,XBOOLEAN:9 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b being Function of Y,BOOLEAN holds (a 'or' b)=I_el(Y) & 'not' a=I_el(Y) implies b=I_el(Y) proof let a,b be Function of Y,BOOLEAN; assume that A1: (a 'or' b)=I_el(Y) and A2: 'not' a=I_el(Y); for x being Element of Y holds b.x=TRUE proof let x be Element of Y; ('not' a).x= TRUE by A2,BVFUNC_1:def 11; then A3: 'not' a.x = TRUE by MARGREL1:def 19; (a 'or' b).x= TRUE by A1,BVFUNC_1:def 11; then a.x 'or' b.x = TRUE by BVFUNC_1:def 4; hence thesis by A3; end; hence thesis by BVFUNC_1:def 11; end; theorem tt: for a,b,c,d being Function of Y,BOOLEAN holds (a 'imp' b)=I_el(Y ) & (c 'imp' d)=I_el(Y) implies (a '&' c) 'imp' (b '&' d)=I_el(Y) proof let a,b,c,d be Function of Y,BOOLEAN; assume that A1: (a 'imp' b)=I_el(Y) and A2: (c 'imp' d)=I_el(Y); for x being Element of Y holds ((a '&' c) 'imp' (b '&' d)).x=TRUE proof let x be Element of Y; (a 'imp' b).x= TRUE by A1,BVFUNC_1:def 11; then A3: 'not' a.x 'or' b.x = TRUE by BVFUNC_1:def 8; (c 'imp' d).x= TRUE by A2,BVFUNC_1:def 11; then A4: 'not' c.x 'or' (d).x = TRUE by BVFUNC_1:def 8; ((a '&' c) 'imp' (b '&' d)).x ='not' (a '&' c).x 'or' (b '&' d).x by BVFUNC_1:def 8 .='not'( a.x '&' c.x) 'or' (b '&' d).x by MARGREL1:def 20 .=('not' a.x 'or' 'not' c.x) 'or' (b.x '&' (d).x) by MARGREL1:def 20 .=(('not' c.x 'or' 'not' a.x) 'or' b.x) '&' (('not' a.x 'or' 'not' c.x) 'or' (d).x) by XBOOLEAN:9 .=('not' c.x 'or' ('not' a.x 'or' b.x)) '&' (('not' a.x 'or' 'not' c.x) 'or' (d).x) .=TRUE '&' ('not' a.x 'or' TRUE) by A3,A4,BINARITH:11 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem Th22: for a,b,c,d being Function of Y,BOOLEAN holds (a 'imp' b)=I_el(Y ) & (c 'imp' d)=I_el(Y) implies (a 'or' c) 'imp' (b 'or' d) =I_el(Y) proof let a,b,c,d be Function of Y,BOOLEAN; assume that A1: (a 'imp' b)=I_el(Y) and A2: (c 'imp' d)=I_el(Y); for x being Element of Y holds ((a 'or' c) 'imp' (b 'or' d)).x=TRUE proof let x be Element of Y; (a 'imp' b).x= TRUE by A1,BVFUNC_1:def 11; then A3: 'not' a.x 'or' b.x = TRUE by BVFUNC_1:def 8; (c 'imp' d).x= TRUE by A2,BVFUNC_1:def 11; then A4: 'not' c.x 'or' (d).x = TRUE by BVFUNC_1:def 8; ((a 'or' c) 'imp' (b 'or' d)).x ='not' (a 'or' c).x 'or' (b 'or' d).x by BVFUNC_1:def 8 .='not'( a.x 'or' c.x) 'or' (b 'or' d).x by BVFUNC_1:def 4 .=(b.x 'or' (d).x) 'or' ('not' a.x '&' 'not' c.x) by BVFUNC_1:def 4 .=(((d).x 'or' b.x) 'or' 'not' a.x) '&' ((b.x 'or' (d).x) 'or' 'not' c.x) by XBOOLEAN:9 .=((d).x 'or' (b.x 'or' 'not' a.x)) '&' ((b.x 'or' (d).x) 'or' 'not' c.x) .=TRUE '&' (b.x 'or' TRUE) by A3,A4,BINARITH:11 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b being Function of Y,BOOLEAN holds (a '&' 'not' b) 'imp' 'not' a=I_el(Y) implies (a 'imp' b)=I_el(Y) proof let a,b be Function of Y,BOOLEAN; assume A1: (a '&' 'not' b) 'imp' 'not' a=I_el(Y); for x being Element of Y holds (a 'imp' b).x=TRUE proof let x be Element of Y; ((a '&' 'not' b) 'imp' 'not' a).x=TRUE by A1,BVFUNC_1:def 11; then 'not' (a '&' 'not' b).x 'or' ('not' a).x = TRUE by BVFUNC_1:def 8; then 'not'( a.x '&' ('not' b).x) 'or' ('not' a).x=TRUE by MARGREL1:def 20 ; then ('not' a.x 'or' 'not' 'not' b.x) 'or' ('not' a).x=TRUE by MARGREL1:def 19; then ('not' a.x 'or' b.x) 'or' 'not' a.x=TRUE by MARGREL1:def 19; then b.x 'or' ('not' a.x 'or' 'not' a.x)=TRUE by XBOOLEAN:4; hence thesis by BVFUNC_1:def 8; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b being Function of Y,BOOLEAN holds a 'imp' 'not' b=I_el(Y ) implies b 'imp' 'not' a=I_el(Y) proof let a,b be Function of Y,BOOLEAN; assume A1: a 'imp' 'not' b=I_el(Y); for x being Element of Y holds (b 'imp' 'not' a).x=TRUE proof let x be Element of Y; (a 'imp' 'not' b).x=TRUE by A1,BVFUNC_1:def 11; then ('not' a.x) 'or' ('not' b).x=TRUE by BVFUNC_1:def 8; then A2: 'not' a.x 'or' 'not' b.x=TRUE by MARGREL1:def 19; (b 'imp' 'not' a).x =('not' b.x) 'or' ('not' a).x by BVFUNC_1:def 8 .=TRUE by A2,MARGREL1:def 19; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b being Function of Y,BOOLEAN holds 'not' a 'imp' b=I_el(Y ) implies 'not' b 'imp' a=I_el(Y) proof let a,b be Function of Y,BOOLEAN; assume A1: 'not' a 'imp' b=I_el(Y); for x being Element of Y holds ('not' b 'imp' a).x=TRUE proof let x be Element of Y; ('not' a 'imp' b).x=TRUE by A1,BVFUNC_1:def 11; then ('not' ('not' a).x) 'or' b.x=TRUE by BVFUNC_1:def 8; then A2: 'not' 'not' a.x 'or' b.x=TRUE by MARGREL1:def 19; ('not' b 'imp' a).x =('not' ('not' b).x) 'or' a.x by BVFUNC_1:def 8 .=TRUE by A2,MARGREL1:def 19; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem Th26: for a,b being Function of Y,BOOLEAN holds a 'imp' (a 'or' b)= I_el(Y) proof let a,b be Function of Y,BOOLEAN; for x being Element of Y holds (a 'imp' (a 'or' b)).x=TRUE proof let x be Element of Y; (a 'imp' (a 'or' b)).x ='not' a.x 'or' (a 'or' b).x by BVFUNC_1:def 8 .='not' a.x 'or' (a.x 'or' b.x) by BVFUNC_1:def 4 .=('not' a.x 'or' a.x) 'or' b.x .=TRUE 'or' b.x by XBOOLEAN:102 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b being Function of Y,BOOLEAN holds (a 'or' b) 'imp' ( 'not' a 'imp' b)=I_el(Y) proof let a,b be Function of Y,BOOLEAN; for x being Element of Y holds ((a 'or' b) 'imp' ('not' a 'imp' b)).x= TRUE proof let x be Element of Y; ((a 'or' b) 'imp' ('not' a 'imp' b)).x ='not' (a 'or' b).x 'or' ('not' a 'imp' b).x by BVFUNC_1:def 8 .='not'( a.x 'or' b.x) 'or' ('not' a 'imp' b).x by BVFUNC_1:def 4 .=('not' a.x '&' 'not' b.x) 'or' ('not' ('not' a).x 'or' b.x) by BVFUNC_1:def 8 .=(a.x 'or' b.x) 'or' ('not' a.x '&' 'not' b.x) by MARGREL1:def 19 .=((a.x 'or' b.x) 'or' 'not' a.x) '&' ((a.x 'or' b.x) 'or' 'not' b.x) by XBOOLEAN:9 .=((a.x 'or' 'not' a.x) 'or' b.x) '&' (a.x 'or' (b.x 'or' 'not' b.x)) .=(TRUE 'or' b.x) '&' (a.x 'or' (b.x 'or' 'not' b.x)) by XBOOLEAN:102 .=TRUE '&' (a.x 'or' TRUE) by XBOOLEAN:102 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem Th28: for a,b being Function of Y,BOOLEAN holds 'not'( a 'or' b) 'imp' ('not' a '&' 'not' b)=I_el(Y) proof let a,b be Function of Y,BOOLEAN; for x being Element of Y holds ('not'( a 'or' b) 'imp' ('not' a '&' 'not' b)).x=TRUE proof let x be Element of Y; ('not'( a 'or' b) 'imp' ('not' a '&' 'not' b)).x ='not' ('not'( a 'or' b)).x 'or' ('not' a '&' 'not' b).x by BVFUNC_1:def 8 .=(a 'or' b).x 'or' ('not' a '&' 'not' b).x by MARGREL1:def 19 .=(a.x 'or' b.x) 'or' ('not' a '&' 'not' b).x by BVFUNC_1:def 4 .=(a.x 'or' b.x) 'or' (('not' a).x '&' ('not' b).x) by MARGREL1:def 20 .=(a.x 'or' b.x) 'or' ('not' a.x '&' ('not' b).x) by MARGREL1:def 19 .=(a.x 'or' b.x) 'or' ('not' a.x '&' 'not' b.x) by MARGREL1:def 19 .=((a.x 'or' b.x) 'or' 'not' a.x) '&' ((a.x 'or' b.x) 'or' 'not' b.x) by XBOOLEAN:9 .=((a.x 'or' 'not' a.x) 'or' b.x) '&' (a.x 'or' (b.x 'or' 'not' b.x)) .=(TRUE 'or' b.x) '&' (a.x 'or' (b.x 'or' 'not' b.x)) by XBOOLEAN:102 .=TRUE '&' (a.x 'or' TRUE) by XBOOLEAN:102 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b being Function of Y,BOOLEAN holds ('not' a '&' 'not' b) 'imp' 'not'( a 'or' b)=I_el(Y) proof let a,b be Function of Y,BOOLEAN; thus ('not' a '&' 'not' b) 'imp' 'not'( a 'or' b) = 'not'( a 'or' b) 'imp' 'not' (a 'or' b) by BVFUNC_1:13 .= 'not'( a 'or' b) 'imp' ('not' a '&' 'not' b) by BVFUNC_1:13 .=I_el Y by Th28; end; theorem for a,b being Function of Y,BOOLEAN holds 'not'( a 'or' b) 'imp' 'not' a=I_el(Y) proof let a,b be Function of Y,BOOLEAN; for x being Element of Y holds ('not'( a 'or' b) 'imp' 'not' a).x=TRUE proof let x be Element of Y; ('not'( a 'or' b) 'imp' 'not' a).x ='not' ('not'( a 'or' b)).x 'or' ( 'not' a).x by BVFUNC_1:def 8 .='not' 'not' (a 'or' b).x 'or' ('not' a).x by MARGREL1:def 19 .=(a 'or' b).x 'or' 'not' a.x by MARGREL1:def 19 .=(a.x 'or' b.x) 'or' 'not' a.x by BVFUNC_1:def 4 .=(a.x 'or' 'not' a.x) 'or' b.x .=TRUE 'or' b.x by XBOOLEAN:102 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a being Function of Y,BOOLEAN holds (a 'or' a) 'imp' a=I_el( Y) proof let a be Function of Y,BOOLEAN; for x being Element of Y holds ((a 'or' a) 'imp' a).x=TRUE proof let x be Element of Y; ((a 'or' a) 'imp' a).x ='not' a.x 'or' a.x by BVFUNC_1:def 8 .=TRUE by XBOOLEAN:102; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b being Function of Y,BOOLEAN holds (a '&' 'not' a) 'imp' b=I_el(Y) proof let a,b be Function of Y,BOOLEAN; for x being Element of Y holds ((a '&' 'not' a) 'imp' b).x=TRUE proof let x be Element of Y; ((a '&' 'not' a) 'imp' b).x ='not' (a '&' 'not' a).x 'or' b.x by BVFUNC_1:def 8 .='not'( a.x '&' ('not' a).x) 'or' b.x by MARGREL1:def 20 .=('not' a.x 'or' 'not' 'not' a.x) 'or' b.x by MARGREL1:def 19 .=TRUE 'or' b.x by XBOOLEAN:102 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b being Function of Y,BOOLEAN holds (a 'imp' b) 'imp' ( 'not' a 'or' b)=I_el(Y) proof let a,b be Function of Y,BOOLEAN; for x being Element of Y holds ((a 'imp' b) 'imp' ('not' a 'or' b)).x= TRUE proof let x be Element of Y; ((a 'imp' b) 'imp' ('not' a 'or' b)).x ='not' (a 'imp' b).x 'or' ( 'not' a 'or' b).x by BVFUNC_1:def 8 .='not'( 'not' a.x 'or' b.x) 'or' ('not' a 'or' b).x by BVFUNC_1:def 8 .=('not' 'not' a.x '&' 'not' b.x) 'or' (('not' a).x 'or' b.x) by BVFUNC_1:def 4 .=('not' a.x 'or' b.x) 'or' (a.x '&' 'not' b.x) by MARGREL1:def 19 .=(('not' a.x 'or' b.x) 'or' a.x) '&' (('not' a.x 'or' b.x) 'or' 'not' b.x) by XBOOLEAN:9 .=(('not' a.x 'or' a.x) 'or' b.x) '&' ('not' a.x 'or' (b.x 'or' 'not' b.x)) .=(TRUE 'or' b.x) '&' ('not' a.x 'or' (b.x 'or' 'not' b.x)) by XBOOLEAN:102 .=TRUE '&' ('not' a.x 'or' TRUE) by XBOOLEAN:102 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b being Function of Y,BOOLEAN holds (a '&' b) 'imp' 'not'( a 'imp' 'not' b)=I_el(Y) proof let a,b be Function of Y,BOOLEAN; for x being Element of Y holds ((a '&' b) 'imp' 'not'( a 'imp' 'not' b)) .x=TRUE proof let x be Element of Y; ((a '&' b) 'imp' 'not'( a 'imp' 'not' b)).x ='not' (a '&' b).x 'or' ( 'not'( a 'imp' 'not' b)).x by BVFUNC_1:def 8 .='not'( a.x '&' b.x) 'or' ('not'( a 'imp' 'not' b)).x by MARGREL1:def 20 .=('not' a.x 'or' 'not' b.x) 'or' 'not' (a 'imp' 'not' b).x by MARGREL1:def 19 .=('not' a.x 'or' 'not' b.x) 'or' 'not'( 'not' a.x 'or' ('not' b ).x) by BVFUNC_1:def 8 .=('not' a.x 'or' 'not' b.x) 'or' (a.x '&' b.x) by MARGREL1:def 19 .=(('not' a.x 'or' 'not' b.x) 'or' a.x) '&' (('not' a.x 'or' 'not' b.x) 'or' b.x) by XBOOLEAN:9 .=(('not' a.x 'or' a.x) 'or' 'not' b.x) '&' ('not' a.x 'or' ( 'not' b.x 'or' b.x)) .=(TRUE 'or' 'not' b.x) '&' ('not' a.x 'or' ('not' b.x 'or' b. x)) by XBOOLEAN:102 .=TRUE '&' ('not' a.x 'or' TRUE) by XBOOLEAN:102 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b being Function of Y,BOOLEAN holds 'not'( a 'imp' 'not' b ) 'imp' (a '&' b)=I_el(Y) proof let a,b be Function of Y,BOOLEAN; for x being Element of Y holds ('not'( a 'imp' 'not' b) 'imp' (a '&' b)) .x=TRUE proof let x be Element of Y; ('not'( a 'imp' 'not' b) 'imp' (a '&' b)).x ='not' ('not'( a 'imp' 'not' b)).x 'or' (a '&' b).x by BVFUNC_1:def 8 .='not' 'not' (a 'imp' 'not' b).x 'or' (a '&' b).x by MARGREL1:def 19 .='not' 'not' (a 'imp' 'not' b).x 'or' (a.x '&' b.x) by MARGREL1:def 20 .=('not' a.x 'or' ('not' b).x) 'or' (a.x '&' b.x) by BVFUNC_1:def 8 .=('not' a.x 'or' 'not' b.x) 'or' (a.x '&' b.x) by MARGREL1:def 19 .=(('not' a.x 'or' 'not' b.x) 'or' a.x) '&' (('not' a.x 'or' 'not' b.x) 'or' b.x) by XBOOLEAN:9 .=(('not' a.x 'or' a.x) 'or' 'not' b.x) '&' ('not' a.x 'or' ( 'not' b.x 'or' b.x)) .=(TRUE 'or' 'not' b.x) '&' ('not' a.x 'or' ('not' b.x 'or' b. x)) by XBOOLEAN:102 .=TRUE '&' ('not' a.x 'or' TRUE) by XBOOLEAN:102 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem Th36: for a,b being Function of Y,BOOLEAN holds 'not'( a '&' b) 'imp' ('not' a 'or' 'not' b)=I_el(Y) proof let a,b be Function of Y,BOOLEAN; for x being Element of Y holds ('not'( a '&' b) 'imp' ('not' a 'or' 'not' b)).x=TRUE proof let x be Element of Y; ('not'( a '&' b) 'imp' ('not' a 'or' 'not' b)).x ='not' ('not'( a '&' b)).x 'or' ('not' a 'or' 'not' b).x by BVFUNC_1:def 8 .=(a '&' b).x 'or' ('not' a 'or' 'not' b).x by MARGREL1:def 19 .=(a.x '&' b.x) 'or' ('not' a 'or' 'not' b).x by MARGREL1:def 20 .=(('not' a).x 'or' ('not' b).x) 'or' (a.x '&' b.x) by BVFUNC_1:def 4 .=('not' a.x 'or' ('not' b).x) 'or' (a.x '&' b.x) by MARGREL1:def 19 .=('not' a.x 'or' 'not' b.x) 'or' (a.x '&' b.x) by MARGREL1:def 19 .=(('not' a.x 'or' 'not' b.x) 'or' a.x) '&' (('not' a.x 'or' 'not' b.x) 'or' b.x) by XBOOLEAN:9 .=(('not' a.x 'or' a.x) 'or' 'not' b.x) '&' ('not' a.x 'or' ( 'not' b.x 'or' b.x)) .=(TRUE 'or' 'not' b.x) '&' ('not' a.x 'or' ('not' b.x 'or' b. x)) by XBOOLEAN:102 .=TRUE '&' ('not' a.x 'or' TRUE) by XBOOLEAN:102 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b being Function of Y,BOOLEAN holds ('not' a 'or' 'not' b) 'imp' 'not'( a '&' b)=I_el(Y) proof let a,b be Function of Y,BOOLEAN; thus ('not' a 'or' 'not' b) 'imp' 'not'( a '&' b) = 'not'( a '&' b) 'imp' 'not' (a '&' b) by BVFUNC_1:14 .= 'not'( a '&' b) 'imp' ('not' a 'or' 'not' b) by BVFUNC_1:14 .=I_el Y by Th36; end; theorem Th38: for a,b being Function of Y,BOOLEAN holds (a '&' b) 'imp' a=I_el (Y) proof let a,b be Function of Y,BOOLEAN; for x being Element of Y holds ((a '&' b) 'imp' a).x=TRUE proof let x be Element of Y; ((a '&' b) 'imp' a).x ='not' (a '&' b).x 'or' a.x by BVFUNC_1:def 8 .=('not' a.x 'or' 'not' b.x) 'or' a.x by MARGREL1:def 20 .=('not' a.x 'or' a.x) 'or' 'not' b.x .=TRUE 'or' 'not' b.x by XBOOLEAN:102 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b being Function of Y,BOOLEAN holds (a '&' b) 'imp' (a 'or' b)=I_el(Y) proof let a,b be Function of Y,BOOLEAN; for x being Element of Y holds ((a '&' b) 'imp' (a 'or' b)).x=TRUE proof let x be Element of Y; ((a '&' b) 'imp' (a 'or' b)).x ='not' (a '&' b).x 'or' (a 'or' b).x by BVFUNC_1:def 8 .=('not' a.x 'or' 'not' b.x) 'or' (a 'or' b).x by MARGREL1:def 20 .=('not' a.x 'or' 'not' b.x) 'or' (a.x 'or' b.x) by BVFUNC_1:def 4 .=('not' b.x 'or' ('not' a.x 'or' a.x)) 'or' b.x .=('not' b.x 'or' TRUE) 'or' b.x by XBOOLEAN:102 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b being Function of Y,BOOLEAN holds (a '&' b) 'imp' b=I_el (Y) proof let a,b be Function of Y,BOOLEAN; for x being Element of Y holds ((a '&' b) 'imp' b).x=TRUE proof let x be Element of Y; ((a '&' b) 'imp' b).x ='not' (a '&' b).x 'or' b.x by BVFUNC_1:def 8 .=('not' a.x 'or' 'not' b.x) 'or' b.x by MARGREL1:def 20 .='not' a.x 'or' ('not' b.x 'or' b.x) .='not' a.x 'or' TRUE by XBOOLEAN:102 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a being Function of Y,BOOLEAN holds a 'imp' a '&' a=I_el(Y) proof let a be Function of Y,BOOLEAN; for x being Element of Y holds (a 'imp' (a '&' a)).x=TRUE proof let x be Element of Y; (a 'imp' a '&' a).x =TRUE '&' ('not' a.x 'or' a.x) by BVFUNC_1:def 8 .=TRUE by XBOOLEAN:102; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem Th42: for a,b being Function of Y,BOOLEAN holds (a 'eqv' b) 'imp' (a 'imp' b)=I_el(Y) proof let a,b be Function of Y,BOOLEAN; for x being Element of Y holds ((a 'eqv' b) 'imp' (a 'imp' b)).x=TRUE proof let x be Element of Y; ((a 'eqv' b) 'imp' (a 'imp' b)).x ='not' (a 'eqv' b).x 'or' (a 'imp' b ).x by BVFUNC_1:def 8 .=(a.x 'xor' b.x) 'or' (a 'imp' b).x by BVFUNC_1:def 9 .=(('not' a.x '&' b.x) 'or' (a.x '&' 'not' b.x)) 'or' 'not'( ( a).x '&' 'not' b.x) by BVFUNC_1:def 8 .=('not' a.x '&' b.x) 'or' ((a.x '&' 'not' b.x) 'or' 'not'( (a ).x '&' 'not' b.x)) .=('not' a.x '&' b.x) 'or' TRUE by XBOOLEAN:102 .=TRUE; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b being Function of Y,BOOLEAN holds (a 'eqv' b) 'imp' (b 'imp' a)=I_el(Y) by Th42; theorem for a,b,c being Function of Y,BOOLEAN holds ((a 'or' b) 'or' c) 'imp' (a 'or' (b 'or' c))=I_el(Y) proof let a,b,c be Function of Y,BOOLEAN; for x being Element of Y holds (((a 'or' b) 'or' c) 'imp' (a 'or' (b 'or' c))).x=TRUE proof let x be Element of Y; (((a 'or' b) 'or' c) 'imp' (a 'or' (b 'or' c))).x ='not' ((a 'or' b) 'or' c).x 'or' (a 'or' (b 'or' c)).x by BVFUNC_1:def 8 .='not'( (a 'or' b).x 'or' c.x) 'or' (a 'or' (b 'or' c)).x by BVFUNC_1:def 4 .='not'( (a.x 'or' b.x) 'or' c.x) 'or' (a 'or' (b 'or' c)).x by BVFUNC_1:def 4 .='not'( (a.x 'or' b.x) 'or' c.x) 'or' (a.x 'or' (b 'or' c).x) by BVFUNC_1:def 4 .='not'( (a.x 'or' b.x) 'or' c.x) 'or' (a.x 'or' (b.x 'or' ( c).x)) by BVFUNC_1:def 4 .=TRUE by XBOOLEAN:102; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b,c being Function of Y,BOOLEAN holds ((a '&' b) '&' c) 'imp' (a '&' (b '&' c))=I_el(Y) proof let a,b,c be Function of Y,BOOLEAN; for x being Element of Y holds (((a '&' b) '&' c) 'imp' (a '&' (b '&' c) )).x=TRUE proof let x be Element of Y; (((a '&' b) '&' c) 'imp' (a '&' (b '&' c))).x ='not' ((a '&' b) '&' c) .x 'or' (a '&' (b '&' c)).x by BVFUNC_1:def 8 .='not'( (a '&' b).x '&' c.x) 'or' (a '&' (b '&' c)).x by MARGREL1:def 20 .='not'( (a.x '&' b.x) '&' c.x) 'or' (a '&' (b '&' c)).x by MARGREL1:def 20 .='not'( (a.x '&' b.x) '&' c.x) 'or' (a.x '&' (b '&' c).x) by MARGREL1:def 20 .='not'( (a.x '&' b.x) '&' c.x) 'or' (a.x '&' (b.x '&' c.x )) by MARGREL1:def 20 .=TRUE by XBOOLEAN:102; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'or' (b 'or' c)) 'imp' ((a 'or' b) 'or' c)=I_el(Y) proof let a,b,c be Function of Y,BOOLEAN; for x being Element of Y holds ((a 'or' (b 'or' c)) 'imp' ((a 'or' b) 'or' c)).x=TRUE proof let x be Element of Y; ((a 'or' (b 'or' c)) 'imp' ((a 'or' b) 'or' c)).x ='not' (a 'or' (b 'or' c)).x 'or' ((a 'or' b) 'or' c).x by BVFUNC_1:def 8 .='not'( a.x 'or' (b 'or' c).x) 'or' ((a 'or' b) 'or' c).x by BVFUNC_1:def 4 .='not'( a.x 'or' (b.x 'or' c.x)) 'or' ((a 'or' b) 'or' c).x by BVFUNC_1:def 4 .='not'( a.x 'or' (b.x 'or' c.x)) 'or' ((a 'or' b).x 'or' c.x) by BVFUNC_1:def 4 .='not'( a.x 'or' (b.x 'or' c.x)) 'or' ((a.x 'or' b.x) 'or' c.x) by BVFUNC_1:def 4 .=TRUE by XBOOLEAN:102; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; begin :: BVFUNC_7 reserve Y for non empty set; theorem for a,b being Function of Y,BOOLEAN holds (a 'imp' b) '&' ('not' a 'imp' b) = b proof let a,b be Function of Y,BOOLEAN; let x be Element of Y; A1: ((a 'imp' b) '&' ('not' a 'imp' b)).x =(a 'imp' b).x '&' ('not' a 'imp' b).x by MARGREL1:def 20 .=('not' a.x 'or' b.x) '&' ('not' a 'imp' b).x by BVFUNC_1:def 8 .=('not' a.x 'or' b.x) '&' ('not' ('not' a).x 'or' b.x) by BVFUNC_1:def 8 .=('not' a.x 'or' b.x) '&' (a.x 'or' b.x) by MARGREL1:def 19; now per cases by XBOOLEAN:def 3; case a.x=TRUE; then ((a 'imp' b) '&' ('not' a 'imp' b)).x =(FALSE 'or' b.x) '&' ( TRUE 'or' b.x) by A1 .=(FALSE 'or' b.x) '&' TRUE .=TRUE '&' b.x .=b.x; hence thesis; end; case a.x=FALSE; then ((a 'imp' b) '&' ('not' a 'imp' b)).x =(TRUE 'or' b.x) '&' ( FALSE 'or' b.x) by A1 .=TRUE '&' (FALSE 'or' b.x) .=TRUE '&' b.x .=b.x; hence thesis; end; end; hence thesis; end; theorem for a,b being Function of Y,BOOLEAN holds (a 'imp' b) '&' (a 'imp' 'not' b) = 'not' a proof let a,b be Function of Y,BOOLEAN; let x be Element of Y; A1: ((a 'imp' b) '&' (a 'imp' 'not' b)).x =(a 'imp' b).x '&' (a 'imp' 'not' b).x by MARGREL1:def 20 .=('not' a.x 'or' b.x) '&' (a 'imp' 'not' b).x by BVFUNC_1:def 8 .=('not' a.x 'or' b.x) '&' ('not' a.x 'or' ('not' b).x) by BVFUNC_1:def 8 .=('not' a.x 'or' b.x) '&' ('not' a.x 'or' 'not' b.x) by MARGREL1:def 19; now per cases by XBOOLEAN:def 3; case b.x=TRUE; then ((a 'imp' b) '&' (a 'imp' 'not' b)).x =('not' a.x 'or' TRUE) '&' ('not' a.x 'or' FALSE) by A1 .=('not' a.x 'or' TRUE) '&' 'not' a.x .=TRUE '&' 'not' a.x .='not' a.x .=('not' a).x by MARGREL1:def 19; hence thesis; end; case b.x=FALSE; then ((a 'imp' b) '&' (a 'imp' 'not' b)).x =('not' a.x 'or' FALSE) '&' ('not' a.x 'or' TRUE) by A1 .='not' a.x '&' ('not' a.x 'or' TRUE) .=TRUE '&' 'not' a.x .='not' a.x .=('not' a).x by MARGREL1:def 19; hence thesis; end; end; hence thesis; end; theorem Th73: for a,b,c being Function of Y,BOOLEAN holds a 'imp' (b 'or' c) = (a 'imp' b) 'or' (a 'imp' c) proof let a,b,c be Function of Y,BOOLEAN; let x be Element of Y; ((a 'imp' b) 'or' (a 'imp' c)).x =(a 'imp' b).x 'or' (a 'imp' c).x by BVFUNC_1:def 4 .=('not' a.x 'or' b.x) 'or' (a 'imp' c).x by BVFUNC_1:def 8 .=('not' a.x 'or' b.x) 'or' ('not' a.x 'or' c.x) by BVFUNC_1:def 8 .=('not' a.x 'or' ('not' a.x 'or' b.x)) 'or' c.x .=(('not' a.x 'or' 'not' a.x) 'or' b.x) 'or' c.x by BINARITH:11 .='not' a.x 'or' (b.x 'or' c.x) .='not' a.x 'or' (b 'or' c).x by BVFUNC_1:def 4 .=(a 'imp' (b 'or' c)).x by BVFUNC_1:def 8; hence thesis; end; theorem for a,b,c being Function of Y,BOOLEAN holds a 'imp' (b '&' c) = (a 'imp' b) '&' (a 'imp' c) proof let a,b,c be Function of Y,BOOLEAN; let x be Element of Y; ((a 'imp' b) '&' (a 'imp' c)).x =(a 'imp' b).x '&' (a 'imp' c).x by MARGREL1:def 20 .=('not' a.x 'or' b.x) '&' (a 'imp' c).x by BVFUNC_1:def 8 .=('not' a.x 'or' b.x) '&' ('not' a.x 'or' c.x) by BVFUNC_1:def 8 .='not' a.x 'or' (b.x '&' c.x) by XBOOLEAN:9 .='not' a.x 'or' ((b '&' c).x) by MARGREL1:def 20 .=(a 'imp' (b '&' c)).x by BVFUNC_1:def 8; hence thesis; end; theorem Th75: for a,b,c being Function of Y,BOOLEAN holds (a 'or' b) 'imp' c = (a 'imp' c) '&' (b 'imp' c) proof let a,b,c be Function of Y,BOOLEAN; let x be Element of Y; ((a 'imp' c) '&' (b 'imp' c)).x =(a 'imp' c).x '&' (b 'imp' c).x by MARGREL1:def 20 .=('not' a.x 'or' c.x) '&' (b 'imp' c).x by BVFUNC_1:def 8 .=(c.x 'or' 'not' a.x) '&' ('not' b.x 'or' c.x) by BVFUNC_1:def 8 .='not'( a.x 'or' b.x) 'or' c.x by XBOOLEAN:9 .='not' (a 'or' b).x 'or' c.x by BVFUNC_1:def 4 .=((a 'or' b) 'imp' c).x by BVFUNC_1:def 8; hence thesis; end; theorem Th76: for a,b,c being Function of Y,BOOLEAN holds (a '&' b) 'imp' c = (a 'imp' c) 'or' (b 'imp' c) proof let a,b,c be Function of Y,BOOLEAN; let x be Element of Y; ((a 'imp' c) 'or' (b 'imp' c)).x =(a 'imp' c).x 'or' (b 'imp' c).x by BVFUNC_1:def 4 .=('not' a.x 'or' c.x) 'or' (b 'imp' c).x by BVFUNC_1:def 8 .=('not' a.x 'or' c.x) 'or' ('not' b.x 'or' c.x) by BVFUNC_1:def 8 .=('not' a.x 'or' (c.x 'or' 'not' b.x)) 'or' c.x .=(('not' a.x 'or' 'not' b.x) 'or' c.x) 'or' c.x .=('not' a.x 'or' 'not' b.x) 'or' (c.x 'or' c.x) by BINARITH:11 .='not' (a '&' b).x 'or' c.x by MARGREL1:def 20 .=((a '&' b) 'imp' c).x by BVFUNC_1:def 8; hence thesis; end; theorem Th7: for a,b,c being Function of Y,BOOLEAN holds (a '&' b) 'imp' c = a 'imp' (b 'imp' c) proof let a,b,c be Function of Y,BOOLEAN; let x be Element of Y; (a 'imp' (b 'imp' c)).x ='not' a.x 'or' (b 'imp' c).x by BVFUNC_1:def 8 .='not' a.x 'or' ('not' b.x 'or' c.x) by BVFUNC_1:def 8 .=('not' a.x 'or' 'not' b.x) 'or' c.x .='not' (a '&' b).x 'or' c.x by MARGREL1:def 20 .=((a '&' b) 'imp' c).x by BVFUNC_1:def 8; hence thesis; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a '&' b) 'imp' c = a 'imp' ('not' b 'or' c) proof let a,b,c be Function of Y,BOOLEAN; let x be Element of Y; (a 'imp' ('not' b 'or' c)).x =(a 'imp' (b 'imp' c)).x by BVFUNC_4:8; hence thesis by Th7; end; theorem for a,b,c being Function of Y,BOOLEAN holds a 'imp' (b 'or' c) = (a '&' 'not' b) 'imp' c proof let a,b,c be Function of Y,BOOLEAN; let x be Element of Y; ((a '&' 'not' b) 'imp' c).x ='not' (a '&' 'not' b).x 'or' c.x by BVFUNC_1:def 8 .=('not' a.x 'or' 'not' ('not' b).x) 'or' c.x by MARGREL1:def 20 .=('not' a.x 'or' b.x) 'or' c.x by MARGREL1:def 19 .='not' a.x 'or' (b.x 'or' c.x) .='not' a.x 'or' (b 'or' c).x by BVFUNC_1:def 4 .=(a 'imp' (b 'or' c)).x by BVFUNC_1:def 8; hence thesis; end; theorem for a,b being Function of Y,BOOLEAN holds a '&' (a 'imp' b) = a '&' b proof let a,b be Function of Y,BOOLEAN; let x be Element of Y; (a '&' (a 'imp' b)).x =a.x '&' (a 'imp' b).x by MARGREL1:def 20 .=a.x '&' ('not' a.x 'or' b.x) by BVFUNC_1:def 8 .=(a.x '&' 'not' a.x) 'or' (a.x '&' b.x) by XBOOLEAN:8 .=FALSE 'or' (a.x '&' b.x) by XBOOLEAN:138 .=a.x '&' b.x .=(a '&' b).x by MARGREL1:def 20; hence thesis; end; theorem for a,b being Function of Y,BOOLEAN holds (a 'imp' b) '&' 'not' b = 'not' a '&' 'not' b proof let a,b be Function of Y,BOOLEAN; let x be Element of Y; ((a 'imp' b) '&' 'not' b).x =(a 'imp' b).x '&' ('not' b).x by MARGREL1:def 20 .=('not' b).x '&' ('not' a.x 'or' b.x) by BVFUNC_1:def 8 .=(('not' b).x '&' 'not' a.x) 'or' (('not' b).x '&' b.x) by XBOOLEAN:8 .=(('not' b).x '&' 'not' a.x) 'or' (b.x '&' 'not' b.x) by MARGREL1:def 19 .=(('not' b).x '&' 'not' a.x) 'or' FALSE by XBOOLEAN:138 .=('not' b).x '&' 'not' a.x .=('not' b).x '&' ('not' a).x by MARGREL1:def 19 .=('not' a '&' 'not' b).x by MARGREL1:def 20; hence thesis; end; theorem Th12: for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) = (a 'imp' b) '&' (b 'imp' c) '&' (a 'imp' c) proof let a,b,c be Function of Y,BOOLEAN; let x be Element of Y; A1: ((a 'imp' b) '&' (b 'imp' c) '&' (a 'imp' c)).x =((a 'imp' b) '&' (b 'imp' c)).x '&' ((a 'imp' c)).x by MARGREL1:def 20 .=((a 'imp' b).x '&' (b 'imp' c).x) '&' ((a 'imp' c)).x by MARGREL1:def 20 .=(('not' a.x 'or' b.x) '&' (b 'imp' c).x) '&' ((a 'imp' c)).x by BVFUNC_1:def 8 .=(('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x)) '&' ((a 'imp' c)).x by BVFUNC_1:def 8 .=(('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x)) '&' ('not' ( a).x 'or' c.x) by BVFUNC_1:def 8 .=(('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x)) '&' 'not' (a ).x 'or' (('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x)) '&' c.x by XBOOLEAN:8; A2: (('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x)) =((a 'imp' b). x '&' ('not' b.x 'or' c.x)) by BVFUNC_1:def 8 .=(a 'imp' b).x '&' (b 'imp' c).x by BVFUNC_1:def 8 .=((a 'imp' b) '&' (b 'imp' c)).x by MARGREL1:def 20; A3: ((a 'imp' b) '&' (b 'imp' c)).x =(a 'imp' b).x '&' (b 'imp' c).x by MARGREL1:def 20 .=('not' a.x 'or' b.x) '&' (b 'imp' c).x by BVFUNC_1:def 8 .=('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x) by BVFUNC_1:def 8 ; now per cases by XBOOLEAN:def 3; case a.x=TRUE & c.x=TRUE; then ((a 'imp' b) '&' (b 'imp' c) '&' (a 'imp' c)).x =(('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x)) '&' FALSE 'or' (('not' a.x 'or' (b) .x) '&' ('not' b.x 'or' c.x)) '&' TRUE by A1 .=FALSE 'or' (('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x )) '&' TRUE .=FALSE 'or' (('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x )) .=((a 'imp' b) '&' (b 'imp' c)).x by A2; hence thesis; end; case A4: a.x=TRUE & c.x=FALSE; then A5: ((a 'imp' b) '&' (b 'imp' c)).x =(FALSE 'or' b.x) '&' ('not' (b ).x 'or' FALSE) by A3 .=(FALSE 'or' b.x) '&' 'not' b.x .=b.x '&' 'not' b.x .=FALSE by XBOOLEAN:138; ((a 'imp' b) '&' (b 'imp' c) '&' (a 'imp' c)).x =(('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x)) '&' FALSE 'or' (('not' a.x 'or' (b) .x) '&' ('not' b.x 'or' c.x)) '&' FALSE by A1,A4 .=FALSE; hence thesis by A5; end; case a.x=FALSE & c.x=TRUE; then ((a 'imp' b) '&' (b 'imp' c) '&' (a 'imp' c)).x =(('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x)) '&' TRUE 'or' (('not' a.x 'or' b. x) '&' ('not' b.x 'or' c.x)) '&' TRUE by A1 .=((a 'imp' b) '&' (b 'imp' c)).x by A2; hence thesis; end; case a.x=FALSE & c.x=FALSE; then ((a 'imp' b) '&' (b 'imp' c) '&' (a 'imp' c)).x =TRUE '&' (('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x)) 'or' FALSE '&' (('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x)) by A1 .=(('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x)) 'or' FALSE '&' (('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x)) .=(('not' a.x 'or' b.x) '&' ('not' b.x 'or' c.x)) 'or' FALSE .=((a 'imp' b) '&' (b 'imp' c)).x by A2; hence thesis; end; end; hence thesis; end; theorem for a being Function of Y,BOOLEAN holds I_el(Y) 'imp' a = a proof let a be Function of Y,BOOLEAN; let x be Element of Y; (I_el(Y) 'imp' a).x ='not' (I_el(Y)).x 'or' a.x by BVFUNC_1:def 8 .=FALSE 'or' a.x by BVFUNC_1:def 11; hence thesis; end; theorem for a being Function of Y,BOOLEAN holds a 'imp' O_el(Y) = 'not' a proof let a be Function of Y,BOOLEAN; let x be Element of Y; (a 'imp' O_el(Y)).x ='not' a.x 'or' (O_el(Y)).x by BVFUNC_1:def 8 .='not' a.x 'or' FALSE by BVFUNC_1:def 10 .='not' a.x; hence thesis by MARGREL1:def 19; end; theorem for a being Function of Y,BOOLEAN holds O_el(Y) 'imp' a = I_el(Y ) proof let a be Function of Y,BOOLEAN; for x being Element of Y holds (O_el(Y) 'imp' a).x = TRUE proof let x be Element of Y; (O_el(Y) 'imp' a).x ='not' (O_el(Y)).x 'or' a.x by BVFUNC_1:def 8 .=TRUE 'or' a.x by BVFUNC_1:def 10; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a being Function of Y,BOOLEAN holds a 'imp' I_el(Y) = I_el(Y ) proof let a be Function of Y,BOOLEAN; for x being Element of Y holds (a 'imp' I_el(Y)).x = TRUE proof let x be Element of Y; (a 'imp' I_el(Y)).x ='not' a.x 'or' (I_el(Y)).x by BVFUNC_1:def 8 .='not' a.x 'or' TRUE by BVFUNC_1:def 11; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a being Function of Y,BOOLEAN holds a 'imp' 'not' a = 'not' a proof let a be Function of Y,BOOLEAN; let x be Element of Y; (a 'imp' 'not' a).x ='not' a.x 'or' ('not' a).x by BVFUNC_1:def 8 .=('not' a).x 'or' ('not' a).x by MARGREL1:def 19 .=('not' a).x; hence thesis; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '<' (c 'imp' a) 'imp' (c 'imp' b) proof let a,b,c be Function of Y,BOOLEAN; let z be Element of Y; assume (a 'imp' b).z=TRUE; then A1: 'not' a.z 'or' b.z = TRUE by BVFUNC_1:def 8; A2: b.z=TRUE or b.z=FALSE by XBOOLEAN:def 3; now per cases by A1,A2; case A3: 'not' a.z=TRUE; ((c 'imp' a) 'imp' (c 'imp' b)).z ='not' (c 'imp' a).z 'or' (c 'imp' b).z by BVFUNC_1:def 8 .='not'( 'not' c.z 'or' a.z) 'or' (c 'imp' b).z by BVFUNC_1:def 8 .=c.z 'or' (c 'imp' b).z by A3 .=c.z 'or' ('not' c.z 'or' b.z) by BVFUNC_1:def 8 .=(c.z 'or' 'not' c.z) 'or' b.z .=TRUE 'or' b.z by XBOOLEAN:102 .=TRUE; hence thesis; end; case A4: b.z=TRUE; ((c 'imp' a) 'imp' (c 'imp' b)).z ='not' (c 'imp' a).z 'or' (c 'imp' b).z by BVFUNC_1:def 8 .='not'( 'not' c.z 'or' a.z) 'or' (c 'imp' b).z by BVFUNC_1:def 8 .='not'( 'not' c.z 'or' a.z) 'or' ('not' c.z 'or' TRUE) by A4, BVFUNC_1:def 8 .='not'( 'not' c.z 'or' a.z) 'or' TRUE .=TRUE; hence thesis; end; end; hence thesis; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'eqv' b) '<' (a 'eqv' c) 'eqv' (b 'eqv' c) proof let a,b,c be Function of Y,BOOLEAN; let z be Element of Y; A1: (a 'eqv' b).z =((a 'imp' b) '&' (b 'imp' a)).z by BVFUNC_4:7 .=(a 'imp' b).z '&' (b 'imp' a).z by MARGREL1:def 20; assume A2: (a 'eqv' b).z=TRUE; then (a 'imp' b).z=TRUE by A1,MARGREL1:12; then A3: 'not' a.z 'or' b.z = TRUE by BVFUNC_1:def 8; A4: a.z=TRUE or a.z=FALSE by XBOOLEAN:def 3; A5: b.z=TRUE or b.z=FALSE by XBOOLEAN:def 3; A6: (b 'imp' a).z = 'not' b.z 'or' a.z by BVFUNC_1:def 8; A7: ((a 'eqv' c) 'eqv' (b 'eqv' c)).z =(((a 'eqv' c) 'imp' (b 'eqv' c)) '&' ((b 'eqv' c) 'imp' (a 'eqv' c))).z by BVFUNC_4:7 .=((((a 'imp' c) '&' (c 'imp' a)) 'imp' (b 'eqv' c)) '&' ((b 'eqv' c) 'imp' (a 'eqv' c))).z by BVFUNC_4:7 .=((((a 'imp' c) '&' (c 'imp' a)) 'imp' (b 'imp' c) '&' (c 'imp' b)) '&' ((b 'eqv' c) 'imp' (a 'eqv' c))).z by BVFUNC_4:7 .=((((a 'imp' c) '&' (c 'imp' a)) 'imp' (b 'imp' c) '&' (c 'imp' b)) '&' (((b 'imp' c) '&' (c 'imp' b)) 'imp' (a 'eqv' c))).z by BVFUNC_4:7 .=((((a 'imp' c) '&' (c 'imp' a)) 'imp' ((b 'imp' c) '&' (c 'imp' b))) '&' (((b 'imp' c) '&' (c 'imp' b)) 'imp' ((a 'imp' c) '&' (c 'imp' a)))).z by BVFUNC_4:7 .=(((('not' a 'or' c) '&' (c 'imp' a)) 'imp' ((b 'imp' c) '&' (c 'imp' b ))) '&' (((b 'imp' c) '&' (c 'imp' b)) 'imp' ((a 'imp' c) '&' (c 'imp' a)))).z by BVFUNC_4:8 .=(((('not' a 'or' c) '&' ('not' c 'or' a)) 'imp' ((b 'imp' c) '&' (c 'imp' b))) '&' (((b 'imp' c) '&' (c 'imp' b)) 'imp' ((a 'imp' c) '&' (c 'imp' a )))).z by BVFUNC_4:8 .=(((('not' a 'or' c) '&' ('not' c 'or' a)) 'imp' (('not' b 'or' c) '&' (c 'imp' b))) '&' (((b 'imp' c) '&' (c 'imp' b)) 'imp' ((a 'imp' c) '&' (c 'imp' a)))).z by BVFUNC_4:8 .=(((('not' a 'or' c) '&' ('not' c 'or' a)) 'imp' (('not' b 'or' c) '&' ('not' c 'or' b))) '&' (((b 'imp' c) '&' (c 'imp' b)) 'imp' ((a 'imp' c) '&' (c 'imp' a)))).z by BVFUNC_4:8 .=(((('not' a 'or' c) '&' ('not' c 'or' a)) 'imp' (('not' b 'or' c) '&' ('not' c 'or' b))) '&' ((('not' b 'or' c) '&' (c 'imp' b)) 'imp' ((a 'imp' c) '&' (c 'imp' a)))).z by BVFUNC_4:8 .=(((('not' a 'or' c) '&' ('not' c 'or' a)) 'imp' (('not' b 'or' c) '&' ('not' c 'or' b))) '&' ((('not' b 'or' c) '&' ('not' c 'or' b)) 'imp' ((a 'imp' c) '&' (c 'imp' a)))).z by BVFUNC_4:8 .=(((('not' a 'or' c) '&' ('not' c 'or' a)) 'imp' (('not' b 'or' c) '&' ('not' c 'or' b))) '&' ((('not' b 'or' c) '&' ('not' c 'or' b)) 'imp' (('not' a 'or' c) '&' (c 'imp' a)))).z by BVFUNC_4:8 .=(((('not' a 'or' c) '&' ('not' c 'or' a)) 'imp' (('not' b 'or' c) '&' ('not' c 'or' b))) '&' ((('not' b 'or' c) '&' ('not' c 'or' b)) 'imp' (('not' a 'or' c) '&' ('not' c 'or' a)))).z by BVFUNC_4:8 .=(('not'( ('not' a 'or' c) '&' ('not' c 'or' a)) 'or' (('not' b 'or' c) '&' ('not' c 'or' b))) '&' ((('not' b 'or' c) '&' ('not' c 'or' b)) 'imp' (( 'not' a 'or' c) '&' ('not' c 'or' a)))).z by BVFUNC_4:8 .=(('not'( ('not' a 'or' c) '&' ('not' c 'or' a)) 'or' (('not' b 'or' c) '&' ('not' c 'or' b))) '&' ('not'( ('not' b 'or' c) '&' ('not' c 'or' b)) 'or' (('not' a 'or' c) '&' ('not' c 'or' a)))).z by BVFUNC_4:8 .=((('not'( 'not' a 'or' c) 'or' 'not'( 'not' c 'or' a)) 'or' (('not' b 'or' c) '&' ('not' c 'or' b))) '&' ('not'( ('not' b 'or' c) '&' ('not' c 'or' b )) 'or' (('not' a 'or' c) '&' ('not' c 'or' a)))).z by BVFUNC_1:14 .=((('not'( 'not' a 'or' c) 'or' 'not'( 'not' c 'or' a)) 'or' (('not' b 'or' c) '&' ('not' c 'or' b))) '&' (('not'( 'not' b 'or' c) 'or' 'not'( 'not' c 'or' b)) 'or' (('not' a 'or' c) '&' ('not' c 'or' a)))).z by BVFUNC_1:14 .=(((('not' 'not' a '&' 'not' c) 'or' 'not'( 'not' c 'or' a)) 'or' (( 'not' b 'or' c) '&' ('not' c 'or' b))) '&' (('not'( 'not' b 'or' c) 'or' 'not'( 'not' c 'or' b)) 'or' (('not' a 'or' c) '&' ('not' c 'or' a)))).z by BVFUNC_1:13 .=(((('not' 'not' a '&' 'not' c) 'or' ('not' 'not' c '&' 'not' a)) 'or' (('not' b 'or' c) '&' ('not' c 'or' b))) '&' (('not'( 'not' b 'or' c) 'or' 'not'( 'not' c 'or' b)) 'or' (('not' a 'or' c) '&' ('not' c 'or' a)))).z by BVFUNC_1:13 .=(((('not' 'not' a '&' 'not' c) 'or' ('not' 'not' c '&' 'not' a)) 'or' (('not' b 'or' c) '&' ('not' c 'or' b))) '&' ((('not' 'not' b '&' 'not' c) 'or' 'not'( 'not' c 'or' b)) 'or' (('not' a 'or' c) '&' ('not' c 'or' a)))).z by BVFUNC_1:13 .=((((a '&' 'not' c) 'or' (c '&' 'not' a)) 'or' (('not' b 'or' c) '&' ( 'not' c 'or' b))) '&' (((b '&' 'not' c) 'or' ('not' 'not' c '&' 'not' b)) 'or' (('not' a 'or' c) '&' ('not' c 'or' a)))).z by BVFUNC_1:13 .=(((a '&' 'not' c) 'or' (c '&' 'not' a)) 'or' (('not' b 'or' c) '&' ( 'not' c 'or' b))).z '&' (((b '&' 'not' c) 'or' (c '&' 'not' b)) 'or' (('not' a 'or' c) '&' ('not' c 'or' a))).z by MARGREL1:def 20 .=(((a '&' 'not' c) 'or' (c '&' 'not' a)).z 'or' (('not' b 'or' c) '&' ( 'not' c 'or' b)).z) '&' (((b '&' 'not' c) 'or' (c '&' 'not' b)) 'or' (('not' a 'or' c) '&' ('not' c 'or' a))).z by BVFUNC_1:def 4 .=(((a '&' 'not' c) 'or' (c '&' 'not' a)).z 'or' (('not' b 'or' c) '&' ( 'not' c 'or' b)).z) '&' (((b '&' 'not' c) 'or' (c '&' 'not' b)).z 'or' (('not' a 'or' c) '&' ('not' c 'or' a)).z) by BVFUNC_1:def 4 .=(((a '&' 'not' c).z 'or' (c '&' 'not' a).z) 'or' (('not' b 'or' c) '&' ('not' c 'or' b)).z) '&' (((b '&' 'not' c) 'or' (c '&' 'not' b)).z 'or' (('not' a 'or' c) '&' ('not' c 'or' a)).z) by BVFUNC_1:def 4 .=(((a '&' 'not' c).z 'or' (c '&' 'not' a).z) 'or' (('not' b 'or' c).z '&' ('not' c 'or' b).z)) '&' (((b '&' 'not' c) 'or' (c '&' 'not' b)).z 'or' (( 'not' a 'or' c) '&' ('not' c 'or' a)).z) by MARGREL1:def 20 .=(((a '&' 'not' c).z 'or' (c '&' 'not' a).z) 'or' (('not' b 'or' c).z '&' ('not' c 'or' b).z)) '&' (((b '&' 'not' c).z 'or' (c '&' 'not' b).z) 'or' ( ('not' a 'or' c) '&' ('not' c 'or' a)).z) by BVFUNC_1:def 4 .=(((a '&' 'not' c).z 'or' (c '&' 'not' a).z) 'or' (('not' b 'or' c).z '&' ('not' c 'or' b).z)) '&' (((b '&' 'not' c).z 'or' (c '&' 'not' b).z) 'or' ( ('not' a 'or' c).z '&' ('not' c 'or' a).z)) by MARGREL1:def 20 .=(((a.z '&' ('not' c).z) 'or' (c '&' 'not' a).z) 'or' (('not' b 'or' c).z '&' ('not' c 'or' b).z)) '&' (((b '&' 'not' c).z 'or' (c '&' 'not' b).z) 'or' (('not' a 'or' c).z '&' ('not' c 'or' a).z)) by MARGREL1:def 20 .=(((a.z '&' ('not' c).z) 'or' (c.z '&' ('not' a).z)) 'or' (('not' b 'or' c).z '&' ('not' c 'or' b).z)) '&' (((b '&' 'not' c).z 'or' (c '&' 'not' b) .z) 'or' (('not' a 'or' c).z '&' ('not' c 'or' a).z)) by MARGREL1:def 20 .=(((a.z '&' ('not' c).z) 'or' (c.z '&' ('not' a).z)) 'or' ((('not' b).z 'or' c.z) '&' ('not' c 'or' b).z)) '&' (((b '&' 'not' c).z 'or' (c '&' 'not' b).z) 'or' (('not' a 'or' c).z '&' ('not' c 'or' a).z)) by BVFUNC_1:def 4 .=(((a.z '&' ('not' c).z) 'or' (c.z '&' ('not' a).z)) 'or' ((('not' b).z 'or' c.z) '&' (('not' c).z 'or' b.z))) '&' (((b '&' 'not' c).z 'or' (c '&' 'not' b).z) 'or' (('not' a 'or' c).z '&' ('not' c 'or' a).z)) by BVFUNC_1:def 4 .=(((a.z '&' ('not' c).z) 'or' (c.z '&' ('not' a).z)) 'or' ((('not' b).z 'or' c.z) '&' (('not' c).z 'or' b.z))) '&' (((b.z '&' ('not' c).z) 'or' (c '&' 'not' b).z) 'or' (('not' a 'or' c).z '&' ('not' c 'or' a).z)) by MARGREL1:def 20 .=(((a.z '&' ('not' c).z) 'or' (c.z '&' ('not' a).z)) 'or' ((('not' b).z 'or' c.z) '&' (('not' c).z 'or' b.z))) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' ('not' b).z)) 'or' (('not' a 'or' c).z '&' ('not' c 'or' a).z)) by MARGREL1:def 20 .=(((a.z '&' ('not' c).z) 'or' (c.z '&' ('not' a).z)) 'or' ((('not' b).z 'or' c.z) '&' (('not' c).z 'or' b.z))) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' ('not' b).z)) 'or' ((('not' a).z 'or' c.z) '&' ('not' c 'or' a).z)) by BVFUNC_1:def 4 .=(((a.z '&' ('not' c).z) 'or' (c.z '&' ('not' a).z)) 'or' ((('not' b).z 'or' c.z) '&' (('not' c).z 'or' b.z))) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' ('not' b).z)) 'or' ((('not' a).z 'or' c.z) '&' (('not' c).z 'or' a.z))) by BVFUNC_1:def 4 .=(((a.z '&' ('not' c).z) 'or' (c.z '&' 'not' a.z)) 'or' ((('not' b).z 'or' c.z) '&' (('not' c).z 'or' b.z))) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' ('not' b).z)) 'or' ((('not' a).z 'or' c.z) '&' (('not' c).z 'or' a.z))) by MARGREL1:def 19 .=(((a.z '&' ('not' c).z) 'or' (c.z '&' 'not' a.z)) 'or' (('not' ( b).z 'or' c.z) '&' (('not' c).z 'or' b.z))) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' ('not' b).z)) 'or' ((('not' a).z 'or' c.z) '&' (('not' c).z 'or' a.z))) by MARGREL1:def 19 .=(((a.z '&' ('not' c).z) 'or' (c.z '&' 'not' a.z)) 'or' (('not' ( b).z 'or' c.z) '&' (('not' c).z 'or' b.z))) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' 'not' b.z)) 'or' ((('not' a).z 'or' c.z) '&' (('not' c).z 'or' a.z))) by MARGREL1:def 19 .=(((a.z '&' ('not' c).z) 'or' (c.z '&' 'not' a.z)) 'or' (('not' ( b).z 'or' c.z) '&' (('not' c).z 'or' b.z))) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' 'not' b.z)) 'or' (('not' a.z 'or' c.z) '&' (('not' c).z 'or' a.z))) by MARGREL1:def 19; (b 'imp' a).z=TRUE by A2,A1,MARGREL1:12; then A8: 'not' b.z=TRUE or a.z=TRUE by A6,A4; now per cases by A3,A5; case A9: 'not' a.z=TRUE; then a.z=FALSE; then ((a 'eqv' c) 'eqv' (b 'eqv' c)).z =((FALSE 'or' (c.z '&' TRUE)) 'or' (('not' b.z 'or' c.z) '&' (('not' c).z 'or' b.z))) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' 'not' b.z)) 'or' ((TRUE 'or' c.z) '&' (('not' c).z 'or' FALSE))) by A7 .=((FALSE 'or' c.z) 'or' (('not' b.z 'or' c.z) '&' (('not' c). z 'or' b.z))) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' 'not' b.z)) 'or' ((TRUE 'or' c.z) '&' (('not' c).z 'or' FALSE))) .=(c.z 'or' (('not' b.z 'or' c.z) '&' (('not' c).z 'or' b.z) )) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' 'not' b.z)) 'or' ((TRUE 'or' c.z) '&' (('not' c).z 'or' FALSE))) .=(c.z 'or' (('not' b.z 'or' c.z) '&' (('not' c).z 'or' b.z) )) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' 'not' b.z)) 'or' ((TRUE 'or' c.z) '&' ('not' c).z)) .=(c.z 'or' (('not' b.z 'or' c.z) '&' (('not' c).z 'or' b.z) )) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' 'not' b.z)) 'or' (TRUE '&' ( 'not' c).z)) .=(c.z 'or' (('not' b.z 'or' c.z) '&' (('not' c).z 'or' b.z) )) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' 'not' b.z)) 'or' ('not' c).z ) .=((c.z 'or' (c.z 'or' 'not' b.z)) '&' (c.z 'or' (('not' c). z 'or' b.z))) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' 'not' b.z)) 'or' ('not' c).z) by XBOOLEAN:9 .=(((c.z 'or' c.z) 'or' 'not' b.z) '&' (c.z 'or' (('not' c). z 'or' b.z))) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' 'not' b.z)) 'or' ('not' c).z) by BINARITH:11 .=((c.z 'or' 'not' b.z) '&' ((c.z 'or' ('not' c).z) 'or' b.z )) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' 'not' b.z)) 'or' ('not' c).z ) .=((c.z 'or' 'not' b.z) '&' ((c.z 'or' 'not' c.z) 'or' b.z )) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' 'not' b.z)) 'or' ('not' c).z ) by MARGREL1:def 19 .=((c.z 'or' 'not' b.z) '&' (TRUE 'or' b.z)) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' 'not' b.z)) 'or' ('not' c).z) by XBOOLEAN:102 .=(TRUE '&' (c.z 'or' 'not' b.z)) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' 'not' b.z)) 'or' ('not' c).z) .=(c.z 'or' 'not' b.z) '&' (((b.z '&' ('not' c).z) 'or' (c.z '&' 'not' b.z)) 'or' ('not' c).z) .=(c.z 'or' 'not' b.z) '&' ((b.z '&' ('not' c).z) 'or' (('not' c).z 'or' (c.z '&' 'not' b.z))) .=(c.z 'or' 'not' b.z) '&' ((b.z '&' ('not' c).z) 'or' ((( 'not' c).z 'or' c.z) '&' (('not' c).z 'or' 'not' b.z))) by XBOOLEAN:9 .=(c.z 'or' 'not' b.z) '&' ((b.z '&' ('not' c).z) 'or' (('not' c.z 'or' c.z) '&' (('not' c).z 'or' 'not' b.z))) by MARGREL1:def 19 .=(c.z 'or' 'not' b.z) '&' ((b.z '&' ('not' c).z) 'or' (TRUE '&' (('not' c).z 'or' 'not' b.z))) by XBOOLEAN:102 .=(c.z 'or' 'not' b.z) '&' ((('not' c).z 'or' 'not' b.z) 'or' (b.z '&' ('not' c).z)) .=(c.z 'or' 'not' b.z) '&' (((('not' c).z 'or' 'not' b.z) 'or' b.z) '&' ((('not' c).z 'or' 'not' b.z) 'or' ('not' c).z)) by XBOOLEAN:9 .=(c.z 'or' 'not' b.z) '&' ((('not' c).z 'or' ('not' b.z 'or' b.z)) '&' ((('not' c).z 'or' 'not' b.z) 'or' ('not' c).z)) .=(c.z 'or' 'not' b.z) '&' ((('not' c).z 'or' TRUE) '&' ((('not' c).z 'or' 'not' b.z) 'or' ('not' c).z)) by XBOOLEAN:102 .=(c.z 'or' 'not' b.z) '&' (TRUE '&' ((('not' c).z 'or' 'not' (b ).z) 'or' ('not' c).z)) .=(c.z 'or' 'not' b.z) '&' (('not' b.z 'or' ('not' c).z) 'or' ('not' c).z) .=(c.z 'or' 'not' b.z) '&' ('not' b.z 'or' (('not' c).z 'or' ( 'not' c).z)) by BINARITH:11 .=('not' b.z '&' (c.z 'or' 'not' b.z)) 'or' ((c.z 'or' 'not' b.z) '&' ('not' c).z) by XBOOLEAN:8 .=(('not' b.z '&' c.z) 'or' ('not' b.z '&' 'not' b.z)) 'or' (('not' c).z '&' (c.z 'or' 'not' b.z)) by XBOOLEAN:8 .=(('not' b.z '&' c.z) 'or' 'not' b.z) 'or' (('not' c).z '&' ( c).z 'or' ('not' c).z '&' 'not' b.z) by XBOOLEAN:8 .=(('not' b.z '&' c.z) 'or' 'not' b.z) 'or' ((c.z '&' 'not' c.z) 'or' (('not' c).z '&' 'not' b.z)) by MARGREL1:def 19 .=(('not' b.z '&' c.z) 'or' 'not' b.z) 'or' (FALSE 'or' (( 'not' c).z '&' 'not' b.z)) by XBOOLEAN:138 .=('not' b.z 'or' ('not' b.z '&' c.z)) 'or' (('not' c).z '&' 'not' b.z) .='not' b.z 'or' (('not' b.z '&' c.z) 'or' (('not' c).z '&' 'not' b.z)) .=TRUE by A8,A9; hence thesis; end; case b.z=TRUE; then 'not' b.z=FALSE; then ((a 'eqv' c) 'eqv' (b 'eqv' c)).z =((('not' c).z 'or' (FALSE '&' (c ).z)) 'or' ((FALSE 'or' c.z) '&' (('not' c).z 'or' TRUE))) '&' ((('not' c).z 'or' (c.z '&' FALSE)) 'or' ((FALSE 'or' c.z) '&' (('not' c).z 'or' TRUE))) by A2,A1,A6,A4,A7 .=((('not' c).z 'or' FALSE) 'or' ((FALSE 'or' c.z) '&' (('not' c). z 'or' TRUE))) '&' ((('not' c).z 'or' FALSE) 'or' ((FALSE 'or' c.z) '&' (( 'not' c).z 'or' TRUE))) .=((('not' c).z 'or' FALSE) 'or' ((FALSE 'or' c.z) '&' TRUE)) '&' ((('not' c).z 'or' FALSE) 'or' ((FALSE 'or' c.z) '&' TRUE)) .=((('not' c).z 'or' FALSE) 'or' (c.z '&' TRUE)) '&' ((('not' c).z 'or' FALSE) 'or' (c.z '&' TRUE)) .=(('not' c).z 'or' (TRUE '&' c.z)) '&' (('not' c).z 'or' (c.z '&' TRUE)) .=(('not' c).z 'or' c.z) '&' (('not' c).z 'or' c.z) .=('not' c.z 'or' c.z) '&' ('not' c.z 'or' c.z) by MARGREL1:def 19 .=TRUE by XBOOLEAN:102; hence thesis; end; end; hence thesis; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'eqv' b) '<' (a 'imp' c) 'eqv' (b 'imp' c) proof let a,b,c be Function of Y,BOOLEAN; let z be Element of Y; A1: (a 'eqv' b).z =((a 'imp' b) '&' (b 'imp' a)).z by BVFUNC_4:7 .=(a 'imp' b).z '&' (b 'imp' a).z by MARGREL1:def 20; assume A2: (a 'eqv' b).z=TRUE; then (a 'imp' b).z=TRUE by A1,MARGREL1:12; then A3: 'not' a.z 'or' b.z = TRUE by BVFUNC_1:def 8; (b 'imp' a).z=TRUE by A2,A1,MARGREL1:12; then A4: 'not' b.z 'or' a.z = TRUE by BVFUNC_1:def 8; A5: ((a 'imp' c) 'eqv' (b 'imp' c)).z =(((a 'imp' c) 'imp' (b 'imp' c)) '&' ((b 'imp' c) 'imp' (a 'imp' c))).z by BVFUNC_4:7 .=((a 'imp' c) 'imp' (b 'imp' c)).z '&' ((b 'imp' c) 'imp' (a 'imp' c)). z by MARGREL1:def 20 .=('not' (a 'imp' c).z 'or' (b 'imp' c).z) '&' ((b 'imp' c) 'imp' (a 'imp' c)).z by BVFUNC_1:def 8 .=('not' (a 'imp' c).z 'or' (b 'imp' c).z) '&' ('not' (b 'imp' c).z 'or' (a 'imp' c).z) by BVFUNC_1:def 8 .=('not'( 'not' a.z 'or' c.z) 'or' (b 'imp' c).z) '&' ('not' (b 'imp' c).z 'or' (a 'imp' c).z) by BVFUNC_1:def 8 .=('not'( 'not' a.z 'or' c.z) 'or' ('not' b.z 'or' c.z)) '&' ( 'not' (b 'imp' c).z 'or' (a 'imp' c).z) by BVFUNC_1:def 8 .=('not'( 'not' a.z 'or' c.z) 'or' ('not' b.z 'or' c.z)) '&' ( 'not'( 'not' b.z 'or' c.z) 'or' (a 'imp' c).z) by BVFUNC_1:def 8 .=((a.z '&' 'not' c.z) 'or' ('not' b.z 'or' c.z)) '&' ((b.z '&' 'not' c.z) 'or' ('not' a.z 'or' c.z)) by BVFUNC_1:def 8; A6: a.z=TRUE or a.z=FALSE by XBOOLEAN:def 3; A7: (b 'imp' a).z = 'not' b.z 'or' a.z by BVFUNC_1:def 8; A8: b.z=TRUE or b.z=FALSE by XBOOLEAN:def 3; now per cases by A3,A8; case A9: 'not' a.z=TRUE; then a.z=FALSE; then 'not' b.z=TRUE by A4; then ((a 'imp' c) 'eqv' (b 'imp' c)).z =(FALSE 'or' (TRUE 'or' c.z)) '&' (FALSE 'or' (TRUE 'or' c.z)) by A5,A9 .=(TRUE 'or' c.z) '&' (TRUE 'or' c.z) .=TRUE; hence thesis; end; case b.z=TRUE; then 'not' b.z=FALSE; then ((a 'imp' c) 'eqv' (b 'imp' c)).z =('not' c.z 'or' (FALSE 'or' (c ).z)) '&' ('not' c.z 'or' (FALSE 'or' c.z)) by A2,A1,A7,A6,A5 .=('not' c.z 'or' c.z) '&' ('not' c.z 'or' c.z) .=TRUE by XBOOLEAN:102; hence thesis; end; end; hence thesis; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'eqv' b) '<' (c 'imp' a) 'eqv' (c 'imp' b) proof let a,b,c be Function of Y,BOOLEAN; let z be Element of Y; A1: (a 'eqv' b).z =((a 'imp' b) '&' (b 'imp' a)).z by BVFUNC_4:7 .=(a 'imp' b).z '&' (b 'imp' a).z by MARGREL1:def 20; assume A2: (a 'eqv' b).z=TRUE; then (a 'imp' b).z=TRUE by A1,MARGREL1:12; then A3: 'not' a.z 'or' b.z = TRUE by BVFUNC_1:def 8; (b 'imp' a).z=TRUE by A2,A1,MARGREL1:12; then A4: 'not' b.z 'or' a.z = TRUE by BVFUNC_1:def 8; A5: ((c 'imp' a) 'eqv' (c 'imp' b)).z =(((c 'imp' a) 'imp' (c 'imp' b)) '&' ((c 'imp' b) 'imp' (c 'imp' a))).z by BVFUNC_4:7 .=((c 'imp' a) 'imp' (c 'imp' b)).z '&' ((c 'imp' b) 'imp' (c 'imp' a)). z by MARGREL1:def 20 .=('not' (c 'imp' a).z 'or' (c 'imp' b).z) '&' ((c 'imp' b) 'imp' (c 'imp' a)).z by BVFUNC_1:def 8 .=('not' (c 'imp' a).z 'or' (c 'imp' b).z) '&' ('not' (c 'imp' b).z 'or' (c 'imp' a).z) by BVFUNC_1:def 8 .=('not'( 'not' c.z 'or' a.z) 'or' (c 'imp' b).z) '&' ('not' (c 'imp' b).z 'or' (c 'imp' a).z) by BVFUNC_1:def 8 .=('not'( 'not' c.z 'or' a.z) 'or' ('not' c.z 'or' b.z)) '&' ( 'not' (c 'imp' b).z 'or' (c 'imp' a).z) by BVFUNC_1:def 8 .=('not'( 'not' c.z 'or' a.z) 'or' ('not' c.z 'or' b.z)) '&' ( 'not'( 'not' c.z 'or' b.z) 'or' (c 'imp' a).z) by BVFUNC_1:def 8 .=((c.z '&' 'not' a.z) 'or' ('not' c.z 'or' b.z)) '&' ((c.z '&' 'not' b.z) 'or' ('not' c.z 'or' a.z)) by BVFUNC_1:def 8; A7: (b 'imp' a).z = 'not' b.z 'or' a.z by BVFUNC_1:def 8; A8: b.z=TRUE or b.z=FALSE by XBOOLEAN:def 3; now per cases by A3,A8; case A9: 'not' a.z=TRUE; then a.z=FALSE; then 'not' b.z=TRUE by A4; then ((c 'imp' a) 'eqv' (c 'imp' b)).z =(c.z 'or' ('not' c.z 'or' FALSE)) '&' (c.z 'or' ('not' c.z 'or' FALSE)) by A5,A9 .=(c.z 'or' 'not' c.z) '&' (c.z 'or' 'not' c.z) .=TRUE by XBOOLEAN:102; hence thesis; end; case b.z=TRUE; then 'not' b.z=FALSE; then ((c 'imp' a) 'eqv' (c 'imp' b)).z =(FALSE 'or' ('not' c.z 'or' TRUE)) '&' (FALSE 'or' ('not' c.z 'or' TRUE)) by A2,A1,A7,A5, MARGREL1:12 .=('not' c.z 'or' TRUE) '&' ('not' c.z 'or' TRUE) .=TRUE; hence thesis; end; end; hence thesis; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'eqv' b) '<' (a '&' c) 'eqv' (b '&' c) proof let a,b,c be Function of Y,BOOLEAN; let z be Element of Y; A1: (a 'eqv' b).z =((a 'imp' b) '&' (b 'imp' a)).z by BVFUNC_4:7 .=(a 'imp' b).z '&' (b 'imp' a).z by MARGREL1:def 20; assume A2: (a 'eqv' b).z=TRUE; then (a 'imp' b).z=TRUE by A1,MARGREL1:12; then A3: 'not' a.z 'or' b.z = TRUE by BVFUNC_1:def 8; (b 'imp' a).z=TRUE by A2,A1,MARGREL1:12; then A4: 'not' b.z 'or' a.z = TRUE by BVFUNC_1:def 8; A5: ((a '&' c) 'eqv' (b '&' c)).z =(((a '&' c) 'imp' (b '&' c)) '&' ((b '&' c) 'imp' (a '&' c))).z by BVFUNC_4:7 .=((a '&' c) 'imp' (b '&' c)).z '&' ((b '&' c) 'imp' (a '&' c)).z by MARGREL1:def 20 .=('not' (a '&' c).z 'or' (b '&' c).z) '&' ((b '&' c) 'imp' (a '&' c)).z by BVFUNC_1:def 8 .=('not' (a '&' c).z 'or' (b '&' c).z) '&' ('not' (b '&' c).z 'or' (a '&' c).z) by BVFUNC_1:def 8 .=('not'( a.z '&' c.z) 'or' (b '&' c).z) '&' ('not' (b '&' c).z 'or' (a '&' c).z) by MARGREL1:def 20 .=('not'( a.z '&' c.z) 'or' (b.z '&' c.z)) '&' ('not' (b '&' c). z 'or' (a '&' c).z) by MARGREL1:def 20 .=('not'( a.z '&' c.z) 'or' (b.z '&' c.z)) '&' ('not'( b.z '&' c.z) 'or' (a '&' c).z) by MARGREL1:def 20 .=(('not' a.z 'or' 'not' c.z) 'or' (b.z '&' c.z)) '&' (('not' (b ).z 'or' 'not' c.z) 'or' (a.z '&' c.z)) by MARGREL1:def 20; A6: a.z=TRUE or a.z=FALSE by XBOOLEAN:def 3; A7: (b 'imp' a).z = 'not' b.z 'or' a.z by BVFUNC_1:def 8; A8: b.z=TRUE or b.z=FALSE by XBOOLEAN:def 3; now per cases by A3,A8; case A9: 'not' a.z=TRUE; then a.z=FALSE; then 'not' b.z=TRUE by A4; then ((a '&' c) 'eqv' (b '&' c)).z =((TRUE 'or' 'not' c.z) 'or' FALSE) '&' ((TRUE 'or' 'not' c.z) 'or' FALSE) by A5,A9 .=(TRUE 'or' 'not' c.z) '&' (TRUE 'or' 'not' c.z) .=TRUE; hence thesis; end; case b.z=TRUE; then 'not' b.z=FALSE; then ((a '&' c) 'eqv' (b '&' c)).z =((FALSE 'or' 'not' c.z) 'or' c.z ) '&' ((FALSE 'or' 'not' c.z) 'or' c.z) by A2,A1,A7,A6,A5 .=('not' c.z 'or' c.z) '&' ('not' c.z 'or' c.z) .=TRUE by XBOOLEAN:102; hence thesis; end; end; hence thesis; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'eqv' b) '<' (a 'or' c) 'eqv' (b 'or' c) proof let a,b,c be Function of Y,BOOLEAN; let z be Element of Y; A1: (a 'eqv' b).z =((a 'imp' b) '&' (b 'imp' a)).z by BVFUNC_4:7 .=(a 'imp' b).z '&' (b 'imp' a).z by MARGREL1:def 20; assume A2: (a 'eqv' b).z=TRUE; then (a 'imp' b).z=TRUE by A1,MARGREL1:12; then A3: 'not' a.z 'or' b.z = TRUE by BVFUNC_1:def 8; (b 'imp' a).z=TRUE by A2,A1,MARGREL1:12; then A4: 'not' b.z 'or' a.z = TRUE by BVFUNC_1:def 8; A5: ((a 'or' c) 'eqv' (b 'or' c)).z =(((a 'or' c) 'imp' (b 'or' c)) '&' ((b 'or' c) 'imp' (a 'or' c))).z by BVFUNC_4:7 .=((a 'or' c) 'imp' (b 'or' c)).z '&' ((b 'or' c) 'imp' (a 'or' c)).z by MARGREL1:def 20 .=('not' (a 'or' c).z 'or' (b 'or' c).z) '&' ((b 'or' c) 'imp' (a 'or' c )).z by BVFUNC_1:def 8 .=('not' (a 'or' c).z 'or' (b 'or' c).z) '&' ('not' (b 'or' c).z 'or' (a 'or' c).z) by BVFUNC_1:def 8 .=('not'( a.z 'or' c.z) 'or' (b 'or' c).z) '&' ('not' (b 'or' c).z 'or' (a 'or' c).z) by BVFUNC_1:def 4 .=('not'( a.z 'or' c.z) 'or' (b.z 'or' c.z)) '&' ('not' (b 'or' c).z 'or' (a 'or' c).z) by BVFUNC_1:def 4 .=('not'( a.z 'or' c.z) 'or' (b.z 'or' c.z)) '&' ('not'( b.z 'or' c.z) 'or' (a 'or' c).z) by BVFUNC_1:def 4 .=(('not' a.z '&' 'not' c.z) 'or' (b.z 'or' c.z)) '&' (('not' (b ).z '&' 'not' c.z) 'or' (a.z 'or' c.z)) by BVFUNC_1:def 4; A6: a.z=TRUE or a.z=FALSE by XBOOLEAN:def 3; A7: (b 'imp' a).z = 'not' b.z 'or' a.z by BVFUNC_1:def 8; A8: b.z=TRUE or b.z=FALSE by XBOOLEAN:def 3; now per cases by A3,A8; case A9: 'not' a.z=TRUE; then a.z=FALSE; then 'not' b.z=TRUE by A4; then ((a 'or' c) 'eqv' (b 'or' c)).z =('not' c.z 'or' (FALSE 'or' c. z)) '&' ('not' c.z 'or' (FALSE 'or' c.z)) by A5,A9 .=('not' c.z 'or' c.z) '&' ('not' c.z 'or' c.z) .=TRUE by XBOOLEAN:102; hence thesis; end; case b.z=TRUE; then 'not' b.z=FALSE; then ((a 'or' c) 'eqv' (b 'or' c)).z =(FALSE 'or' (TRUE 'or' c.z)) '&' (FALSE 'or' (TRUE 'or' c.z)) by A2,A1,A7,A6,A5 .=(TRUE 'or' c.z) '&' (TRUE 'or' c.z) .=TRUE; hence thesis; end; end; hence thesis; end; theorem for a,b being Function of Y,BOOLEAN holds a '<' (a 'eqv' b) 'eqv' (b 'eqv' a) 'eqv' a proof let a,b be Function of Y,BOOLEAN; let z be Element of Y; A1: ('not' a).z='not' a.z by MARGREL1:def 19; assume A2: a.z=TRUE; then A3: 'not' a.z=FALSE; A4: ((a 'eqv' b) 'eqv' (b 'eqv' a)).z =(((a 'eqv' b) 'imp' (b 'eqv' a)) '&' ((b 'eqv' a) 'imp' (a 'eqv' b))).z by BVFUNC_4:7 .=('not'( a 'eqv' b) 'or' (b 'eqv' a)).z '&' ('not'( b 'eqv' a) 'or' (a 'eqv' b)).z by BVFUNC_4:8 .=(('not'( a 'eqv' b)).z 'or' (b 'eqv' a).z) '&' (('not'( b 'eqv' a)).z 'or' (a 'eqv' b).z) by BVFUNC_1:def 4 .=(('not'( (a 'imp' b) '&' (b 'imp' a))).z 'or' (b 'eqv' a).z) '&' (( 'not'( b 'eqv' a)).z 'or' (a 'eqv' b).z) by BVFUNC_4:7 .=(('not'( (a 'imp' b) '&' (b 'imp' a))).z 'or' (b 'eqv' a).z) '&' (( 'not'( b 'eqv' a)).z 'or' ((a 'imp' b) '&' (b 'imp' a)).z) by BVFUNC_4:7 .=(('not'( (a 'imp' b) '&' (b 'imp' a))).z 'or' ((b 'imp' a) '&' (a 'imp' b)).z) '&' (('not'( b 'eqv' a)).z 'or' ((a 'imp' b) '&' (b 'imp' a)).z) by BVFUNC_4:7 .=(('not'( (a 'imp' b) '&' (b 'imp' a))).z 'or' ((b 'imp' a) '&' (a 'imp' b)).z) '&' (('not'( (b 'imp' a) '&' (a 'imp' b))).z 'or' ((a 'imp' b) '&' (b 'imp' a)).z) by BVFUNC_4:7 .=('not' ((a 'imp' b) '&' (b 'imp' a)).z 'or' ((b 'imp' a) '&' (a 'imp' b)).z) '&' ('not' ((b 'imp' a) '&' (a 'imp' b)).z 'or' ((a 'imp' b) '&' (b 'imp' a)).z) by MARGREL1:def 19 .=('not'( (a 'imp' b).z '&' (b 'imp' a).z) 'or' ((b 'imp' a) '&' (a 'imp' b)).z) '&' ('not' ((b 'imp' a) '&' (a 'imp' b)).z 'or' ((a 'imp' b) '&' ( b 'imp' a)).z) by MARGREL1:def 20 .=('not'( (a 'imp' b).z '&' (b 'imp' a).z) 'or' ((b 'imp' a).z '&' (a 'imp' b).z)) '&' ('not' ((b 'imp' a) '&' (a 'imp' b)).z 'or' ((a 'imp' b) '&' ( b 'imp' a)).z) by MARGREL1:def 20 .=('not'( (a 'imp' b).z '&' (b 'imp' a).z) 'or' ((b 'imp' a).z '&' (a 'imp' b).z)) '&' ('not'( (b 'imp' a).z '&' (a 'imp' b).z) 'or' ((a 'imp' b) '&' (b 'imp' a)).z) by MARGREL1:def 20 .=('not'( (a 'imp' b).z '&' (b 'imp' a).z) 'or' ((b 'imp' a).z '&' (a 'imp' b).z)) '&' ('not'( (b 'imp' a).z '&' (a 'imp' b).z) 'or' ((a 'imp' b).z '&' (b 'imp' a).z)) by MARGREL1:def 20 .=('not'( ('not' a 'or' b).z '&' (b 'imp' a).z) 'or' ((b 'imp' a).z '&' (a 'imp' b).z)) '&' ('not'( (b 'imp' a).z '&' (a 'imp' b).z) 'or' ((a 'imp' b). z '&' (b 'imp' a).z)) by BVFUNC_4:8 .=('not'( ('not' a 'or' b).z '&' ('not' b 'or' a).z) 'or' ((b 'imp' a).z '&' (a 'imp' b).z)) '&' ('not'( (b 'imp' a).z '&' (a 'imp' b).z) 'or' ((a 'imp' b).z '&' (b 'imp' a).z)) by BVFUNC_4:8 .=('not'( ('not' a 'or' b).z '&' ('not' b 'or' a).z) 'or' (('not' b 'or' a).z '&' (a 'imp' b).z)) '&' ('not'( (b 'imp' a).z '&' (a 'imp' b).z) 'or' ((a 'imp' b).z '&' (b 'imp' a).z)) by BVFUNC_4:8 .=('not'( ('not' a 'or' b).z '&' ('not' b 'or' a).z) 'or' (('not' b 'or' a).z '&' ('not' a 'or' b).z)) '&' ('not'( (b 'imp' a).z '&' (a 'imp' b).z) 'or' ((a 'imp' b).z '&' (b 'imp' a).z)) by BVFUNC_4:8 .=('not'( ('not' a 'or' b).z '&' ('not' b 'or' a).z) 'or' (('not' b 'or' a).z '&' ('not' a 'or' b).z)) '&' ('not'( ('not' b 'or' a).z '&' (a 'imp' b).z) 'or' ((a 'imp' b).z '&' (b 'imp' a).z)) by BVFUNC_4:8 .=('not'( ('not' a 'or' b).z '&' ('not' b 'or' a).z) 'or' (('not' b 'or' a).z '&' ('not' a 'or' b).z)) '&' ('not'( ('not' b 'or' a).z '&' ('not' a 'or' b).z) 'or' ((a 'imp' b).z '&' (b 'imp' a).z)) by BVFUNC_4:8 .=('not'( ('not' a 'or' b).z '&' ('not' b 'or' a).z) 'or' (('not' b 'or' a).z '&' ('not' a 'or' b).z)) '&' ('not'( ('not' b 'or' a).z '&' ('not' a 'or' b).z) 'or' (('not' a 'or' b).z '&' (b 'imp' a).z)) by BVFUNC_4:8 .=('not'( ('not' a 'or' b).z '&' ('not' b 'or' a).z) 'or' (('not' b 'or' a).z '&' ('not' a 'or' b).z)) '&' ('not'( ('not' b 'or' a).z '&' ('not' a 'or' b).z) 'or' (('not' a 'or' b).z '&' ('not' b 'or' a).z)) by BVFUNC_4:8 .=('not'( (('not' a).z 'or' b.z) '&' (('not' b 'or' a).z)) 'or' ((( 'not' b 'or' a).z) '&' (('not' a 'or' b).z))) '&' ('not'( (('not' b 'or' a).z) '&' (('not' a 'or' b).z)) 'or' ((('not' a 'or' b).z) '&' (('not' b 'or' a).z))) by BVFUNC_1:def 4 .=('not'( (('not' a).z 'or' b.z) '&' (('not' b).z 'or' a.z)) 'or' (( ('not' b 'or' a).z) '&' (('not' a 'or' b).z))) '&' ('not'( (('not' b 'or' a).z) '&' (('not' a 'or' b).z)) 'or' ((('not' a 'or' b).z) '&' (('not' b 'or' a).z))) by BVFUNC_1:def 4 .=('not'( (('not' a).z 'or' b.z) '&' (('not' b).z 'or' a.z)) 'or' (( ('not' b).z 'or' a.z) '&' (('not' a 'or' b).z))) '&' ('not'( (('not' b 'or' a ).z) '&' (('not' a 'or' b).z)) 'or' ((('not' a 'or' b).z) '&' (('not' b 'or' a) .z))) by BVFUNC_1:def 4 .=('not'( (('not' a).z 'or' b.z) '&' (('not' b).z 'or' a.z)) 'or' (( ('not' b).z 'or' a.z) '&' (('not' a).z 'or' b.z))) '&' ('not'( (('not' b 'or' a).z) '&' (('not' a 'or' b).z)) 'or' ((('not' a 'or' b).z) '&' (('not' b 'or' a).z))) by BVFUNC_1:def 4 .=('not'( (('not' a).z 'or' b.z) '&' (('not' b).z 'or' a.z)) 'or' (( ('not' b).z 'or' a.z) '&' (('not' a).z 'or' b.z))) '&' ('not'( (('not' b).z 'or' a.z) '&' (('not' a 'or' b).z)) 'or' ((('not' a 'or' b).z) '&' (('not' b 'or' a).z))) by BVFUNC_1:def 4 .=('not'( (('not' a).z 'or' b.z) '&' (('not' b).z 'or' a.z)) 'or' (( ('not' b).z 'or' a.z) '&' (('not' a).z 'or' b.z))) '&' ('not'( (('not' b).z 'or' a.z) '&' (('not' a).z 'or' b.z)) 'or' ((('not' a 'or' b).z) '&' (( 'not' b 'or' a).z))) by BVFUNC_1:def 4 .=('not'( (('not' a).z 'or' b.z) '&' (('not' b).z 'or' a.z)) 'or' (( ('not' b).z 'or' a.z) '&' (('not' a).z 'or' b.z))) '&' ('not'( (('not' b).z 'or' a.z) '&' (('not' a).z 'or' b.z)) 'or' ((('not' a).z 'or' b.z) '&' (( 'not' b 'or' a).z))) by BVFUNC_1:def 4 .=('not'( (FALSE 'or' b.z) '&' (('not' b).z 'or' TRUE)) 'or' ((('not' b).z 'or' TRUE) '&' (FALSE 'or' b.z))) '&' ('not'( (('not' b).z 'or' TRUE) '&' (FALSE 'or' b.z)) 'or' ((FALSE 'or' b.z) '&' (('not' b).z 'or' TRUE))) by A3,A1,BVFUNC_1:def 4 .=('not'( b.z '&' (('not' b).z 'or' TRUE)) 'or' ((('not' b).z 'or' TRUE) '&' (FALSE 'or' b.z))) '&' ('not'( (('not' b).z 'or' TRUE) '&' (FALSE 'or' b.z)) 'or' ((FALSE 'or' b.z) '&' (('not' b).z 'or' TRUE))) .=('not'( b.z '&' (('not' b).z 'or' TRUE)) 'or' ((('not' b).z 'or' TRUE) '&' b.z)) '&' ('not'( (('not' b).z 'or' TRUE) '&' (FALSE 'or' b.z)) 'or' ((FALSE 'or' b.z) '&' (('not' b).z 'or' TRUE))) .=('not'( b.z '&' (('not' b).z 'or' TRUE)) 'or' ((('not' b).z 'or' TRUE) '&' b.z)) '&' ('not'( (('not' b).z 'or' TRUE) '&' (FALSE 'or' b.z)) 'or' (b.z '&' (('not' b).z 'or' TRUE))) .=('not'( b.z '&' TRUE) 'or' ((('not' b).z 'or' TRUE) '&' b.z)) '&' ('not'( (('not' b).z 'or' TRUE) '&' (FALSE 'or' b.z)) 'or' (b.z '&' (('not' b).z 'or' TRUE))) .=('not'( b.z '&' TRUE) 'or' (TRUE '&' b.z)) '&' ('not'( (('not' b). z 'or' TRUE) '&' (FALSE 'or' b.z)) 'or' (b.z '&' (('not' b).z 'or' TRUE))) .=('not'( b.z '&' TRUE) 'or' (TRUE '&' b.z)) '&' ('not'( TRUE '&' ( FALSE 'or' b.z)) 'or' (b.z '&' (('not' b).z 'or' TRUE))) .=('not'( b.z '&' TRUE) 'or' (TRUE '&' b.z)) '&' ('not'( TRUE '&' ( FALSE 'or' b.z)) 'or' (b.z '&' TRUE)) .=('not'( TRUE '&' b.z) 'or' (TRUE '&' b.z)) '&' ('not'( TRUE '&' (b ).z) 'or' (b.z '&' TRUE)) .=('not' b.z 'or' (TRUE '&' b.z)) '&' ('not' b.z 'or' (TRUE '&' (b ).z)) .=('not' b.z 'or' b.z) '&' ('not' b.z 'or' b.z) .=TRUE by XBOOLEAN:102; ((a 'eqv' b) 'eqv' (b 'eqv' a) 'eqv' a).z =((((a 'eqv' b) 'eqv' (b 'eqv' a)) 'imp' a) '&' (a 'imp' ((a 'eqv' b) 'eqv' (b 'eqv' a)))).z by BVFUNC_4:7 .=(((a 'eqv' b) 'eqv' (b 'eqv' a)) 'imp' a).z '&' (a 'imp' ((a 'eqv' b) 'eqv' (b 'eqv' a))).z by MARGREL1:def 20 .=('not' ((a 'eqv' b) 'eqv' (b 'eqv' a)).z 'or' a.z) '&' (a 'imp' ((a 'eqv' b) 'eqv' (b 'eqv' a))).z by BVFUNC_1:def 8 .=('not' ((a 'eqv' b) 'eqv' (b 'eqv' a)).z 'or' a.z) '&' ('not' a.z 'or' ((a 'eqv' b) 'eqv' (b 'eqv' a)).z) by BVFUNC_1:def 8 .=(FALSE 'or' a.z) '&' ('not' a.z 'or' TRUE) by A4 .=a.z '&' ('not' a.z 'or' TRUE) .=TRUE '&' a.z .=TRUE by A2; hence thesis; end; theorem for a,b being Function of Y,BOOLEAN holds a '<' (a 'imp' b) 'eqv' b proof let a,b be Function of Y,BOOLEAN; let z be Element of Y; assume a.z=TRUE; then A2: 'not' a.z=FALSE; ((a 'imp' b) 'eqv' b).z =(('not' a 'or' b) 'eqv' b).z by BVFUNC_4:8 .=((('not' a 'or' b) 'imp' b) '&' (b 'imp' ('not' a 'or' b))).z by BVFUNC_4:7 .=(('not'( 'not' a 'or' b) 'or' b) '&' (b 'imp' ('not' a 'or' b))).z by BVFUNC_4:8 .=(('not'( 'not' a 'or' b) 'or' b) '&' ('not' b 'or' ('not' a 'or' b))). z by BVFUNC_4:8 .=('not'( 'not' a 'or' b) 'or' b).z '&' ('not' b 'or' ('not' a 'or' b)). z by MARGREL1:def 20 .=(('not'( 'not' a 'or' b)).z 'or' b.z) '&' ('not' b 'or' ('not' a 'or' b)).z by BVFUNC_1:def 4 .=('not' ('not' a 'or' b).z 'or' b.z) '&' ('not' b 'or' ('not' a 'or' b)).z by MARGREL1:def 19 .=('not'( ('not' a).z 'or' b.z) 'or' b.z) '&' ('not' b 'or' ('not' a 'or' b)).z by BVFUNC_1:def 4 .=(('not' 'not' a.z '&' 'not' b.z) 'or' b.z) '&' ('not' b 'or' ( 'not' a 'or' b)).z by MARGREL1:def 19 .=((a.z '&' 'not' b.z) 'or' b.z) '&' (('not' b).z 'or' ('not' a 'or' b).z) by BVFUNC_1:def 4 .=((a.z '&' 'not' b.z) 'or' b.z) '&' (('not' b).z 'or' (('not' a). z 'or' b.z)) by BVFUNC_1:def 4 .=((a.z '&' 'not' b.z) 'or' b.z) '&' (('not' b).z 'or' ('not' a. z 'or' b.z)) by MARGREL1:def 19 .=((TRUE '&' 'not' b.z) 'or' b.z) '&' ('not' b.z 'or' (FALSE 'or' b.z)) by A2,MARGREL1:def 19 .=('not' b.z 'or' b.z) '&' ('not' b.z 'or' (FALSE 'or' b.z)) .=('not' b.z 'or' b.z) '&' ('not' b.z 'or' b.z) .=TRUE by XBOOLEAN:102; hence thesis; end; theorem for a,b being Function of Y,BOOLEAN holds a '<' (b 'imp' a) 'eqv' a proof let a,b be Function of Y,BOOLEAN; let z be Element of Y; assume a.z=TRUE; then A2: 'not' a.z=FALSE; ((b 'imp' a) 'eqv' a).z =(('not' b 'or' a) 'eqv' a).z by BVFUNC_4:8 .=((('not' b 'or' a) 'imp' a) '&' (a 'imp' ('not' b 'or' a))).z by BVFUNC_4:7 .=(('not'( 'not' b 'or' a) 'or' a) '&' (a 'imp' ('not' b 'or' a))).z by BVFUNC_4:8 .=(('not'( 'not' b 'or' a) 'or' a) '&' ('not' a 'or' ('not' b 'or' a))). z by BVFUNC_4:8 .=('not'( 'not' b 'or' a) 'or' a).z '&' ('not' a 'or' ('not' b 'or' a)). z by MARGREL1:def 20 .=(('not'( 'not' b 'or' a)).z 'or' a.z) '&' ('not' a 'or' ('not' b 'or' a)).z by BVFUNC_1:def 4 .=('not' ('not' b 'or' a).z 'or' a.z) '&' ('not' a 'or' ('not' b 'or' a)).z by MARGREL1:def 19 .=('not'( ('not' b).z 'or' a.z) 'or' a.z) '&' ('not' a 'or' ('not' b 'or' a)).z by BVFUNC_1:def 4 .=(('not' 'not' b.z '&' 'not' a.z) 'or' a.z) '&' ('not' a 'or' ( 'not' b 'or' a)).z by MARGREL1:def 19 .=((b.z '&' 'not' a.z) 'or' a.z) '&' (('not' a).z 'or' ('not' b 'or' a).z) by BVFUNC_1:def 4 .=((b.z '&' 'not' a.z) 'or' a.z) '&' (('not' a).z 'or' (('not' b). z 'or' a.z)) by BVFUNC_1:def 4 .=((b.z '&' 'not' a.z) 'or' a.z) '&' (('not' a).z 'or' ('not' b. z 'or' a.z)) by MARGREL1:def 19 .=((b.z '&' 'not' a.z) 'or' a.z) '&' ('not' a.z 'or' ('not' b. z 'or' a.z)) by MARGREL1:def 19 .=TRUE '&' (FALSE 'or' ('not' b.z 'or' TRUE)) by A2 .=FALSE 'or' ('not' b.z 'or' TRUE) .='not' b.z 'or' TRUE .=TRUE; hence thesis; end; theorem for a,b being Function of Y,BOOLEAN holds a '<' (a '&' b) 'eqv' (b '&' a) 'eqv' a proof let a,b be Function of Y,BOOLEAN; let z be Element of Y; assume A1: a.z=TRUE; A2: ((a '&' b) 'eqv' (a '&' b)).z =(((a '&' b) 'imp' (a '&' b)) '&' ((a '&' b) 'imp' (a '&' b))).z by BVFUNC_4:7 .=('not'( a '&' b) 'or' (a '&' b)).z by BVFUNC_4:8 .=(I_el(Y)).z by BVFUNC_4:6 .=TRUE by BVFUNC_1:def 11; ((a '&' b) 'eqv' (b '&' a) 'eqv' a).z =((((a '&' b) 'eqv' (a '&' b)) 'imp' a) '&' (a 'imp' ((a '&' b) 'eqv' (a '&' b)))).z by BVFUNC_4:7 .=(((a '&' b) 'eqv' (a '&' b)) 'imp' a).z '&' (a 'imp' ((a '&' b) 'eqv' (a '&' b))).z by MARGREL1:def 20 .=('not'( (a '&' b) 'eqv' (a '&' b)) 'or' a).z '&' (a 'imp' ((a '&' b) 'eqv' (a '&' b))).z by BVFUNC_4:8 .=('not'( (a '&' b) 'eqv' (a '&' b)) 'or' a).z '&' ('not' a 'or' ((a '&' b) 'eqv' (a '&' b))).z by BVFUNC_4:8 .=(('not'( (a '&' b) 'eqv' (a '&' b))).z 'or' a.z) '&' ('not' a 'or' ( (a '&' b) 'eqv' (a '&' b))).z by BVFUNC_1:def 4 .=(('not'( (a '&' b) 'eqv' (a '&' b))).z 'or' a.z) '&' (('not' a).z 'or' ((a '&' b) 'eqv' (a '&' b)).z) by BVFUNC_1:def 4 .=('not' ((a '&' b) 'eqv' (a '&' b)).z 'or' a.z) '&' (('not' a).z 'or' ((a '&' b) 'eqv' (a '&' b)).z) by MARGREL1:def 19 .=(FALSE 'or' a.z) '&' (('not' a).z 'or' TRUE) by A2 .=a.z '&' (('not' a).z 'or' TRUE) .=TRUE '&' a.z .=TRUE by A1; hence thesis; end; begin :: BVFUNC_8 reserve Y for non empty set; theorem for a,b,c,d being Function of Y,BOOLEAN holds a 'imp' (b '&' c '&' d) = (a 'imp' b) '&' (a 'imp' c) '&' (a 'imp' d) proof let a,b,c,d be Function of Y,BOOLEAN; let x be Element of Y; ((a 'imp' b) '&' (a 'imp' c) '&' (a 'imp' d)).x =((a 'imp' b) '&' (a 'imp' c)).x '&' (a 'imp' d).x by MARGREL1:def 20 .=(a 'imp' b).x '&' (a 'imp' c).x '&' (a 'imp' d).x by MARGREL1:def 20 .=('not' a.x 'or' b.x) '&' (a 'imp' c).x '&' (a 'imp' d).x by BVFUNC_1:def 8 .=('not' a.x 'or' b.x) '&' ('not' a.x 'or' c.x) '&' (a 'imp' d ).x by BVFUNC_1:def 8 .=('not' a.x 'or' b.x) '&' ('not' a.x 'or' c.x) '&' ('not' (a) .x 'or' (d).x) by BVFUNC_1:def 8 .=('not' a.x 'or' (b.x '&' c.x)) '&' ('not' a.x 'or' (d).x) by XBOOLEAN:9 .='not' a.x 'or' (b.x '&' c.x '&' (d).x) by XBOOLEAN:9 .='not' a.x 'or' ((b '&' c).x '&' (d).x) by MARGREL1:def 20 .='not' a.x 'or' ((b '&' c '&' d).x) by MARGREL1:def 20 .=(a 'imp' (b '&' c '&' d)).x by BVFUNC_1:def 8; hence thesis; end; theorem for a,b,c,d being Function of Y,BOOLEAN holds a 'imp' (b 'or' c 'or' d) = (a 'imp' b) 'or' (a 'imp' c) 'or' (a 'imp' d) proof let a,b,c,d be Function of Y,BOOLEAN; let x be Element of Y; ((a 'imp' b) 'or' (a 'imp' c) 'or' (a 'imp' d)).x =((a 'imp' b) 'or' ( a 'imp' c)).x 'or' (a 'imp' d).x by BVFUNC_1:def 4 .=(a 'imp' b).x 'or' (a 'imp' c).x 'or' (a 'imp' d).x by BVFUNC_1:def 4 .=('not' a.x 'or' b.x) 'or' (a 'imp' c).x 'or' (a 'imp' d).x by BVFUNC_1:def 8 .=('not' a.x 'or' b.x) 'or' ('not' a.x 'or' c.x) 'or' (a 'imp' d).x by BVFUNC_1:def 8 .=('not' a.x 'or' b.x) 'or' ('not' a.x 'or' c.x) 'or' ('not' ( a).x 'or' (d).x) by BVFUNC_1:def 8 .=(('not' a.x 'or' ('not' a.x 'or' b.x)) 'or' c.x) 'or' ('not' a.x 'or' (d).x) .=((('not' a.x 'or' 'not' a.x) 'or' b.x) 'or' c.x) 'or' ('not' a.x 'or' (d).x) by BINARITH:11 .=('not' a.x 'or' (b.x 'or' c.x)) 'or' ('not' a.x 'or' (d).x) .=('not' a.x 'or' (b 'or' c).x) 'or' ('not' a.x 'or' (d).x) by BVFUNC_1:def 4 .=('not' a.x 'or' ('not' a.x 'or' (b 'or' c).x)) 'or' (d).x .=(('not' a.x 'or' 'not' a.x) 'or' (b 'or' c).x) 'or' (d).x by BINARITH:11 .='not' a.x 'or' ((b 'or' c).x 'or' (d).x) .='not' a.x 'or' (b 'or' c 'or' d).x by BVFUNC_1:def 4 .=(a 'imp' (b 'or' c 'or' d)).x by BVFUNC_1:def 8; hence thesis; end; theorem for a,b,c,d being Function of Y,BOOLEAN holds (a '&' b '&' c) 'imp' d = (a 'imp' d) 'or' (b 'imp' d) 'or' (c 'imp' d) proof let a,b,c,d be Function of Y,BOOLEAN; let x be Element of Y; ((a 'imp' d) 'or' (b 'imp' d) 'or' (c 'imp' d)).x =((a 'imp' d) 'or' ( b 'imp' d)).x 'or' (c 'imp' d).x by BVFUNC_1:def 4 .=(a 'imp' d).x 'or' (b 'imp' d).x 'or' (c 'imp' d).x by BVFUNC_1:def 4 .=('not' a.x 'or' (d).x) 'or' (b 'imp' d).x 'or' (c 'imp' d).x by BVFUNC_1:def 8 .=('not' a.x 'or' (d).x) 'or' ('not' b.x 'or' (d).x) 'or' (c 'imp' d).x by BVFUNC_1:def 8 .=('not' a.x 'or' (d).x) 'or' ('not' b.x 'or' (d).x) 'or' ('not' ( c).x 'or' (d).x) by BVFUNC_1:def 8 .=(('not' a.x 'or' ((d).x 'or' 'not' b.x)) 'or' (d).x) 'or' ('not' c.x 'or' (d).x) .=(('not' a.x 'or' 'not' b.x) 'or' (d).x) 'or' (d).x 'or' ('not' ( c).x 'or' (d).x) .=('not' a.x 'or' 'not' b.x) 'or' ((d).x 'or' (d).x) 'or' ('not' ( c).x 'or' (d).x) by BINARITH:11 .=('not'( a.x '&' b.x) 'or' ((d).x 'or' 'not' c.x)) 'or' (d).x .=(('not'( a.x '&' b.x) 'or' 'not' c.x) 'or' (d).x) 'or' (d).x .=('not'( a.x '&' b.x) 'or' 'not' c.x) 'or' ((d).x 'or' (d).x) by BINARITH:11 .='not'( (a '&' b).x '&' c.x) 'or' (d).x by MARGREL1:def 20 .='not' (a '&' b '&' c).x 'or' (d).x by MARGREL1:def 20 .=((a '&' b '&' c) 'imp' d).x by BVFUNC_1:def 8; hence thesis; end; theorem for a,b,c,d being Function of Y,BOOLEAN holds (a 'or' b 'or' c) 'imp' d = (a 'imp' d) '&' (b 'imp' d) '&' (c 'imp' d) proof let a,b,c,d be Function of Y,BOOLEAN; let x be Element of Y; ((a 'imp' d) '&' (b 'imp' d) '&' (c 'imp' d)).x =((a 'imp' d) '&' (b 'imp' d)).x '&' (c 'imp' d).x by MARGREL1:def 20 .=(a 'imp' d).x '&' (b 'imp' d).x '&' (c 'imp' d).x by MARGREL1:def 20 .=('not' a.x 'or' (d).x) '&' (b 'imp' d).x '&' (c 'imp' d).x by BVFUNC_1:def 8 .=('not' a.x 'or' (d).x) '&' ('not' b.x 'or' (d).x) '&' (c 'imp' d ).x by BVFUNC_1:def 8 .=((d).x 'or' 'not' a.x) '&' ('not' b.x 'or' (d).x) '&' ('not' (c) .x 'or' (d).x) by BVFUNC_1:def 8 .=('not'( a.x 'or' b.x) 'or' (d).x) '&' ('not' c.x 'or' (d).x) by XBOOLEAN:9 .=((d).x 'or' 'not' (a 'or' b).x) '&' ('not' c.x 'or' (d).x) by BVFUNC_1:def 4 .=('not'( (a 'or' b).x 'or' c.x)) 'or' (d).x by XBOOLEAN:9 .='not' (a 'or' b 'or' c).x 'or' (d).x by BVFUNC_1:def 4 .=((a 'or' b 'or' c) 'imp' d).x by BVFUNC_1:def 8; hence thesis; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a) = (a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a) '&' (b 'imp' a) '&' (a 'imp' c) proof let a,b,c be Function of Y,BOOLEAN; let x be Element of Y; (a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a) =('not' a 'or' b) '&' (b 'imp' c) '&' (c 'imp' a) by BVFUNC_4:8 .=('not' a 'or' b) '&' ('not' b 'or' c) '&' (c 'imp' a) by BVFUNC_4:8 .=('not' a 'or' b) '&' ('not' b 'or' c) '&' ('not' c 'or' a) by BVFUNC_4:8 .=(('not' a '&' ('not' b 'or' c) 'or' b '&' ('not' b 'or' c))) '&' ( 'not' c 'or' a) by BVFUNC_1:12 .=(('not' a '&' ('not' b 'or' c) 'or' (b '&' 'not' b 'or' b '&' c))) '&' ('not' c 'or' a) by BVFUNC_1:12 .=(('not' a '&' ('not' b 'or' c) 'or' (O_el(Y) 'or' b '&' c))) '&' ( 'not' c 'or' a) by BVFUNC_4:5 .=(('not' a '&' ('not' b 'or' c) 'or' (b '&' c))) '&' ('not' c 'or' a) by BVFUNC_1:9 .=(('not' a 'or' (b '&' c)) '&' (('not' b 'or' c) 'or' (b '&' c))) '&' ('not' c 'or' a) by BVFUNC_1:11 .=((('not' a 'or' b) '&' ('not' a 'or' c)) '&' (('not' b 'or' c) 'or' (b '&' c))) '&' ('not' c 'or' a) by BVFUNC_1:11 .=((('not' a 'or' b) '&' ('not' a 'or' c)) '&' ((('not' b 'or' c) 'or' b) '&' (('not' b 'or' c) 'or' c))) '&' ('not' c 'or' a) by BVFUNC_1:11 .=((('not' a 'or' b) '&' ('not' a 'or' c)) '&' ((c 'or' ('not' b 'or' b)) '&' (('not' b 'or' c) 'or' c))) '&' ('not' c 'or' a) by BVFUNC_1:8 .=((('not' a 'or' b) '&' ('not' a 'or' c)) '&' ((c 'or' I_el(Y)) '&' ( ('not' b 'or' c) 'or' c))) '&' ('not' c 'or' a) by BVFUNC_4:6 .=((('not' a 'or' b) '&' ('not' a 'or' c)) '&' (I_el(Y) '&' (('not' b 'or' c) 'or' c))) '&' ('not' c 'or' a) by BVFUNC_1:10 .=((('not' a 'or' b) '&' ('not' a 'or' c)) '&' (('not' b 'or' c) 'or' c)) '&' ('not' c 'or' a) by BVFUNC_1:6 .=((('not' a 'or' b) '&' ('not' a 'or' c)) '&' ('not' b 'or' (c 'or' c ))) '&' ('not' c 'or' a) by BVFUNC_1:8 .=(('not' a 'or' b) '&' ('not' a 'or' c)) '&' (('not' b 'or' c) '&' ( 'not' c 'or' a)) by BVFUNC_1:4 .=(('not' a 'or' b) '&' ('not' a 'or' c)) '&' (('not' b '&' ('not' c 'or' a)) 'or' (c '&' ('not' c 'or' a))) by BVFUNC_1:12 .=(('not' a 'or' b) '&' ('not' a 'or' c)) '&' (('not' b '&' ('not' c 'or' a)) 'or' ((c '&' 'not' c 'or' c '&' a))) by BVFUNC_1:12 .=(('not' a 'or' b) '&' ('not' a 'or' c)) '&' (('not' b '&' ('not' c 'or' a)) 'or' ((O_el(Y) 'or' c '&' a))) by BVFUNC_4:5 .=(('not' a 'or' b) '&' ('not' a 'or' c)) '&' (('not' b '&' ('not' c 'or' a)) 'or' (c '&' a)) by BVFUNC_1:9 .=(('not' a 'or' b) '&' ('not' a 'or' c)) '&' (('not' b 'or' (c '&' a) ) '&' (('not' c 'or' a) 'or' (c '&' a))) by BVFUNC_1:11 .=(('not' a 'or' b) '&' ('not' a 'or' c)) '&' ((('not' b 'or' c) '&' ( 'not' b 'or' a)) '&' (('not' c 'or' a) 'or' (c '&' a))) by BVFUNC_1:11 .= (('not' a 'or' c) '&' ('not' a 'or' b)) '&' ((('not' b 'or' a) '&' ('not' b 'or' c)) '&' ((('not' c 'or' a) 'or' c) '&' (('not' c 'or' a) 'or' a)) ) by BVFUNC_1:11 .= (('not' a 'or' c) '&' ('not' a 'or' b)) '&' ((('not' b 'or' a) '&' ('not' b 'or' c)) '&' ((a 'or' ('not' c 'or' c)) '&' (('not' c 'or' a) 'or' a)) ) by BVFUNC_1:8 .= (('not' a 'or' c) '&' ('not' a 'or' b)) '&' ((('not' b 'or' a) '&' ('not' b 'or' c)) '&' ((a 'or' I_el(Y)) '&' (('not' c 'or' a) 'or' a))) by BVFUNC_4:6 .= (('not' a 'or' c) '&' ('not' a 'or' b)) '&' ((('not' b 'or' a) '&' ('not' b 'or' c)) '&' (I_el(Y) '&' (('not' c 'or' a) 'or' a))) by BVFUNC_1:10 .= (('not' a 'or' c) '&' ('not' a 'or' b)) '&' ((('not' b 'or' a) '&' ('not' b 'or' c)) '&' (('not' c 'or' a) 'or' a)) by BVFUNC_1:6 .= (('not' a 'or' c) '&' ('not' a 'or' b)) '&' ((('not' b 'or' a) '&' ('not' b 'or' c)) '&' ('not' c 'or' (a 'or' a))) by BVFUNC_1:8 .= ((('not' a 'or' c) '&' ('not' a 'or' b)) '&' (('not' b 'or' a) '&' ('not' b 'or' c))) '&' ('not' c 'or' a) by BVFUNC_1:4 .= (((('not' a 'or' b) '&' ('not' a 'or' c)) '&' ('not' b 'or' a)) '&' ('not' b 'or' c)) '&' ('not' c 'or' a) by BVFUNC_1:4 .= (((('not' b 'or' a) '&' ('not' a 'or' c)) '&' ('not' a 'or' b)) '&' ('not' b 'or' c)) '&' ('not' c 'or' a) by BVFUNC_1:4 .= ((('not' b 'or' a) '&' ('not' a 'or' c)) '&' (('not' a 'or' b) '&' ('not' b 'or' c))) '&' ('not' c 'or' a) by BVFUNC_1:4 .= (('not' b 'or' a) '&' ('not' a 'or' c)) '&' ((('not' a 'or' b) '&' ('not' b 'or' c)) '&' ('not' c 'or' a)) by BVFUNC_1:4 .= ('not' b 'or' a) '&' (('not' a 'or' b) '&' ('not' b 'or' c) '&' ( 'not' c 'or' a)) '&' ('not' a 'or' c) by BVFUNC_1:4 .= (a 'imp' b) '&' ('not' b 'or' c) '&' ('not' c 'or' a) '&' ('not' b 'or' a) '&' ('not' a 'or' c) by BVFUNC_4:8 .= (a 'imp' b) '&' (b 'imp' c) '&' ('not' c 'or' a) '&' ('not' b 'or' a) '&' ('not' a 'or' c) by BVFUNC_4:8 .= (a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a) '&' ('not' b 'or' a) '&' ('not' a 'or' c) by BVFUNC_4:8 .= (a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a) '&' (b 'imp' a) '&' ( 'not' a 'or' c) by BVFUNC_4:8 .=(a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a) '&' (b 'imp' a) '&' (a 'imp' c) by BVFUNC_4:8; hence thesis; end; theorem for a,b being Function of Y,BOOLEAN holds a = (a '&' b) 'or' (a '&' 'not' b) proof let a,b be Function of Y,BOOLEAN; let x be Element of Y; ((a '&' b) 'or' (a '&' 'not' b)).x =(a '&' (b 'or' 'not' b)).x by BVFUNC_1:12 .=(a '&' I_el(Y)).x by BVFUNC_4:6 .=a.x by BVFUNC_1:6; hence thesis; end; theorem for a,b being Function of Y,BOOLEAN holds a = (a 'or' b) '&' (a 'or' 'not' b) proof let a,b be Function of Y,BOOLEAN; let x be Element of Y; ((a 'or' b) '&' (a 'or' 'not' b)).x =(a 'or' (b '&' 'not' b)).x by BVFUNC_1:11 .=(a 'or' O_el(Y)).x by BVFUNC_4:5 .=a.x by BVFUNC_1:9; hence thesis; end; theorem for a,b,c being Function of Y,BOOLEAN holds a = (a '&' b '&' c) 'or' (a '&' b '&' 'not' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' 'not' b '&' 'not' c) proof let a,b,c be Function of Y,BOOLEAN; let x be Element of Y; ((a '&' b '&' c) 'or' (a '&' b '&' 'not' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' 'not' b '&' 'not' c)).x =(((a '&' b) '&' (c 'or' 'not' c)) 'or' (a '&' 'not' b '&' c) 'or' (a '&' 'not' b '&' 'not' c)).x by BVFUNC_1:12 .=(((a '&' b) '&' I_el(Y)) 'or' (a '&' 'not' b '&' c) 'or' (a '&' 'not' b '&' 'not' c)).x by BVFUNC_4:6 .=((a '&' b) 'or' (a '&' 'not' b '&' c) 'or' (a '&' 'not' b '&' 'not' c)).x by BVFUNC_1:6 .=((a '&' b) 'or' ((a '&' 'not' b '&' c) 'or' (a '&' 'not' b '&' 'not' c))).x by BVFUNC_1:8 .=((a '&' b) 'or' ((a '&' 'not' b) '&' (c 'or' 'not' c))).x by BVFUNC_1:12 .=((a '&' b) 'or' ((a '&' 'not' b) '&' I_el(Y))).x by BVFUNC_4:6 .=((a '&' b) 'or' (a '&' 'not' b)).x by BVFUNC_1:6 .=(a '&' (b 'or' 'not' b)).x by BVFUNC_1:12 .=(a '&' I_el(Y)).x by BVFUNC_4:6 .=a.x by BVFUNC_1:6; hence thesis; end; theorem for a,b,c being Function of Y,BOOLEAN holds a = (a 'or' b 'or' c ) '&' (a 'or' b 'or' 'not' c) '&' (a 'or' 'not' b 'or' c) '&' (a 'or' 'not' b 'or' 'not' c) proof let a,b,c be Function of Y,BOOLEAN; let x be Element of Y; ((a 'or' b 'or' c) '&' (a 'or' b 'or' 'not' c) '&' (a 'or' 'not' b 'or' c) '&' (a 'or' 'not' b 'or' 'not' c)).x =(((a 'or' b) 'or' (c '&' 'not' c) ) '&' (a 'or' 'not' b 'or' c) '&' (a 'or' 'not' b 'or' 'not' c)).x by BVFUNC_1:11 .=(((a 'or' b) 'or' O_el(Y)) '&' (a 'or' 'not' b 'or' c) '&' (a 'or' 'not' b 'or' 'not' c)).x by BVFUNC_4:5 .=((a 'or' b) '&' (a 'or' 'not' b 'or' c) '&' (a 'or' 'not' b 'or' 'not' c)).x by BVFUNC_1:9 .=((a 'or' b) '&' ((a 'or' 'not' b 'or' c) '&' (a 'or' 'not' b 'or' 'not' c))).x by BVFUNC_1:4 .=((a 'or' b) '&' ((a 'or' 'not' b) 'or' (c '&' 'not' c))).x by BVFUNC_1:11 .=((a 'or' b) '&' ((a 'or' 'not' b) 'or' O_el(Y))).x by BVFUNC_4:5 .=((a 'or' b) '&' (a 'or' 'not' b)).x by BVFUNC_1:9 .=(a 'or' (b '&' 'not' b)).x by BVFUNC_1:11 .=(a 'or' O_el(Y)).x by BVFUNC_4:5 .=a.x by BVFUNC_1:9; hence thesis; end; theorem for a,b being Function of Y,BOOLEAN holds a '&' b = a '&' ('not' a 'or' b) proof let a,b be Function of Y,BOOLEAN; let x be Element of Y; (a '&' ('not' a 'or' b)).x =a.x '&' ('not' a 'or' b).x by MARGREL1:def 20 .=a.x '&' (('not' a).x 'or' b.x) by BVFUNC_1:def 4 .=a.x '&' ('not' a).x 'or' a.x '&' b.x by XBOOLEAN:8 .=a.x '&' 'not' a.x 'or' a.x '&' b.x by MARGREL1:def 19 .=FALSE 'or' a.x '&' b.x by XBOOLEAN:138 .=a.x '&' b.x .=(a '&' b).x by MARGREL1:def 20; hence thesis; end; theorem for a,b being Function of Y,BOOLEAN holds a 'or' b = a 'or' ( 'not' a '&' b) proof let a,b be Function of Y,BOOLEAN; let x be Element of Y; (a 'or' ('not' a '&' b)).x =a.x 'or' ('not' a '&' b).x by BVFUNC_1:def 4 .=a.x 'or' (('not' a).x '&' b.x) by MARGREL1:def 20 .=(a.x 'or' ('not' a).x) '&' (a.x 'or' b.x) by XBOOLEAN:9 .=(a.x 'or' 'not' a.x) '&' (a.x 'or' b.x) by MARGREL1:def 19 .=TRUE '&' (a.x 'or' b.x) by XBOOLEAN:102 .=a.x 'or' b.x .=(a 'or' b).x by BVFUNC_1:def 4; hence thesis; end; theorem for a,b being Function of Y,BOOLEAN holds a 'xor' b = 'not'( a 'eqv' b) proof let a,b be Function of Y,BOOLEAN; let x be Element of Y; (a 'xor' b).x =('not' 'not'( ('not' a '&' b) 'or' (a '&' 'not' b))).x by BVFUNC_4:9 .=('not'('not'('not' a '&' b) '&' 'not'( a '&' 'not' b))).x by BVFUNC_1:13 .=('not'( ('not' 'not' a 'or' 'not' b) '&' 'not'( a '&' 'not' b))).x by BVFUNC_1:14 .=('not'( (a 'or' 'not' b) '&' ('not' a 'or' 'not' 'not' b))).x by BVFUNC_1:14 .=('not'( (b 'imp' a) '&' ('not' a 'or' b))).x by BVFUNC_4:8 .=('not'( (b 'imp' a) '&' (a 'imp' b))).x by BVFUNC_4:8 .=('not'( a 'eqv' b)).x by BVFUNC_4:7; hence thesis; end; theorem for a,b being Function of Y,BOOLEAN holds a 'xor' b = (a 'or' b) '&' ('not' a 'or' 'not' b) proof let a,b be Function of Y,BOOLEAN; let x be Element of Y; ((a 'or' b) '&' ('not' a 'or' 'not' b)).x =(a 'or' b).x '&' ('not' a 'or' 'not' b).x by MARGREL1:def 20 .=(a.x 'or' b.x) '&' ('not' a 'or' 'not' b).x by BVFUNC_1:def 4 .=(a.x 'or' b.x) '&' (('not' a).x 'or' ('not' b).x) by BVFUNC_1:def 4 .=(('not' a).x '&' (a.x 'or' b.x)) 'or' ((a.x 'or' b.x) '&' ( 'not' b).x) by XBOOLEAN:8 .=(('not' a).x '&' a.x 'or' ('not' a).x '&' b.x) 'or' (('not' b).x '&' (a.x 'or' b.x)) by XBOOLEAN:8 .=(('not' a).x '&' a.x 'or' ('not' a).x '&' b.x) 'or' (('not' b).x '&' a.x 'or' ('not' b).x '&' b.x) by XBOOLEAN:8 .=('not' a.x '&' a.x 'or' ('not' a).x '&' b.x) 'or' (('not' b).x '&' a.x 'or' ('not' b).x '&' b.x) by MARGREL1:def 19 .=('not' a.x '&' a.x 'or' ('not' a).x '&' b.x) 'or' (('not' b).x '&' a.x 'or' 'not' b.x '&' b.x) by MARGREL1:def 19 .=(FALSE 'or' ('not' a).x '&' b.x) 'or' (('not' b).x '&' a.x 'or' 'not' b.x '&' b.x) by XBOOLEAN:138 .=(FALSE 'or' ('not' a).x '&' b.x) 'or' (('not' b).x '&' a.x 'or' FALSE) by XBOOLEAN:138 .=(('not' a).x '&' b.x) 'or' (('not' b).x '&' a.x 'or' FALSE) .=(('not' a).x '&' b.x) 'or' (a.x '&' ('not' b).x) .=('not' a '&' b).x 'or' (a.x '&' ('not' b).x) by MARGREL1:def 20 .=('not' a '&' b).x 'or' (a '&' 'not' b).x by MARGREL1:def 20 .=('not' a '&' b 'or' a '&' 'not' b).x by BVFUNC_1:def 4 .=(a 'xor' b).x by BVFUNC_4:9; hence thesis; end; theorem for a being Function of Y,BOOLEAN holds a 'xor' I_el(Y) = 'not' a proof let a be Function of Y,BOOLEAN; let x be Element of Y; (a 'xor' I_el(Y)).x =(('not' a '&' I_el(Y)) 'or' (a '&' 'not' I_el(Y)) ).x by BVFUNC_4:9 .=(('not' a '&' I_el(Y)) 'or' (a '&' O_el(Y))).x by BVFUNC_1:2 .=(('not' a '&' I_el(Y)) 'or' O_el(Y)).x by BVFUNC_1:5 .=('not' a '&' I_el(Y)).x by BVFUNC_1:9 .=('not' a).x by BVFUNC_1:6; hence thesis; end; theorem for a being Function of Y,BOOLEAN holds a 'xor' O_el(Y) = a proof let a be Function of Y,BOOLEAN; let x be Element of Y; (a 'xor' O_el(Y)).x =(('not' a '&' O_el(Y)) 'or' (a '&' 'not' O_el(Y)) ).x by BVFUNC_4:9 .=(('not' a '&' O_el(Y)) 'or' (a '&' I_el(Y))).x by BVFUNC_1:2 .=(('not' a '&' O_el(Y)) 'or' a).x by BVFUNC_1:6 .=(O_el(Y) 'or' a).x by BVFUNC_1:5 .=a.x by BVFUNC_1:9; hence thesis; end; theorem for a,b being Function of Y,BOOLEAN holds a 'xor' b = 'not' a 'xor' 'not' b proof let a,b be Function of Y,BOOLEAN; let x be Element of Y; ('not' a 'xor' 'not' b).x =(('not' 'not' a '&' 'not' b) 'or' ('not' a '&' 'not' 'not' b)).x by BVFUNC_4:9 .=(a 'xor' b).x by BVFUNC_4:9; hence thesis; end; theorem for a,b being Function of Y,BOOLEAN holds 'not'( a 'xor' b) = a 'xor' 'not' b proof let a,b be Function of Y,BOOLEAN; let x be Element of Y; ('not'( a 'xor' b)).x =('not'( ('not' a '&' b) 'or' (a '&' 'not' b))). x by BVFUNC_4:9 .=(('not'('not' a '&' b) '&' 'not'( a '&' 'not' b))).x by BVFUNC_1:13 .=((('not' 'not' a 'or' 'not' b) '&' 'not'( a '&' 'not' b))).x by BVFUNC_1:14 .=(((a 'or' 'not' b) '&' ('not' a 'or' 'not' 'not' b))).x by BVFUNC_1:14 .=(((a 'or' 'not' b) '&' 'not' a 'or' (a 'or' 'not' b) '&' b)).x by BVFUNC_1:12 .=(((a '&' 'not' a 'or' 'not' b '&' 'not' a) 'or' (a 'or' 'not' b) '&' b)).x by BVFUNC_1:12 .=(((O_el(Y) 'or' 'not' b '&' 'not' a) 'or' (a 'or' 'not' b) '&' b)).x by BVFUNC_4:5 .=((('not' b '&' 'not' a) 'or' (a 'or' 'not' b) '&' b)).x by BVFUNC_1:9 .=((('not' b '&' 'not' a) 'or' (a '&' b 'or' 'not' b '&' b))).x by BVFUNC_1:12 .=((('not' b '&' 'not' a) 'or' (a '&' b 'or' O_el(Y)))).x by BVFUNC_4:5 .=(('not' a '&' 'not' b) 'or' (a '&' 'not' 'not' b)).x by BVFUNC_1:9 .=(a 'xor' 'not' b).x by BVFUNC_4:9; hence thesis; end; theorem Th18: for a,b being Function of Y,BOOLEAN holds a 'eqv' b = (a 'or' 'not' b) '&' ('not' a 'or' b) proof let a,b be Function of Y,BOOLEAN; let x be Element of Y; ((a 'or' 'not' b) '&' ('not' a 'or' b)).x =((a 'or' 'not' b) '&' (a 'imp' b)).x by BVFUNC_4:8 .=((a 'imp' b) '&' (b 'imp' a)).x by BVFUNC_4:8 .=(a 'eqv' b).x by BVFUNC_4:7; hence thesis; end; theorem for a,b being Function of Y,BOOLEAN holds a 'eqv' b = (a '&' b) 'or' ('not' a '&' 'not' b) proof let a,b be Function of Y,BOOLEAN; let x be Element of Y; ((a '&' b) 'or' ('not' a '&' 'not' b)).x =(((a '&' b) 'or' 'not' a) '&' ((a '&' b) 'or' 'not' b)).x by BVFUNC_1:11 .=(((a 'or' 'not' a) '&' (b 'or' 'not' a)) '&' ((a '&' b) 'or' 'not' b )).x by BVFUNC_1:11 .=(((a 'or' 'not' a) '&' (b 'or' 'not' a)) '&' ((a 'or' 'not' b) '&' ( b 'or' 'not' b))).x by BVFUNC_1:11 .=((I_el(Y) '&' (b 'or' 'not' a)) '&' ((a 'or' 'not' b) '&' (b 'or' 'not' b))).x by BVFUNC_4:6 .=((I_el(Y) '&' (b 'or' 'not' a)) '&' ((a 'or' 'not' b) '&' I_el(Y))). x by BVFUNC_4:6 .=((b 'or' 'not' a) '&' ((a 'or' 'not' b) '&' I_el(Y))).x by BVFUNC_1:6 .=((b 'or' 'not' a) '&' (a 'or' 'not' b)).x by BVFUNC_1:6 .=(a 'eqv' b).x by Th18; hence thesis; end; theorem for a being Function of Y,BOOLEAN holds a 'eqv' I_el(Y) = a proof let a be Function of Y,BOOLEAN; let x be Element of Y; (a 'eqv' I_el(Y)).x =((a 'imp' I_el(Y)) '&' (I_el(Y) 'imp' a)).x by BVFUNC_4:7 .=(('not' a 'or' I_el(Y)) '&' (I_el(Y) 'imp' a)).x by BVFUNC_4:8 .=(('not' a 'or' I_el(Y)) '&' ('not' I_el(Y) 'or' a)).x by BVFUNC_4:8 .=(I_el(Y) '&' ('not' I_el(Y) 'or' a)).x by BVFUNC_1:10 .=(I_el(Y) '&' (O_el(Y) 'or' a)).x by BVFUNC_1:2 .=(I_el(Y) '&' a).x by BVFUNC_1:9 .=a.x by BVFUNC_1:6; hence thesis; end; theorem for a being Function of Y,BOOLEAN holds a 'eqv' O_el(Y) = 'not' a proof let a be Function of Y,BOOLEAN; let x be Element of Y; (a 'eqv' O_el(Y)).x =((a 'imp' O_el(Y)) '&' (O_el(Y) 'imp' a)).x by BVFUNC_4:7 .=(('not' a 'or' O_el(Y)) '&' (O_el(Y) 'imp' a)).x by BVFUNC_4:8 .=(('not' a 'or' O_el(Y)) '&' ('not' O_el(Y) 'or' a)).x by BVFUNC_4:8 .=('not' a '&' ('not' O_el(Y) 'or' a)).x by BVFUNC_1:9 .=('not' a '&' (I_el(Y) 'or' a)).x by BVFUNC_1:2 .=('not' a '&' I_el(Y)).x by BVFUNC_1:10 .=('not' a).x by BVFUNC_1:6; hence thesis; end; theorem for a,b being Function of Y,BOOLEAN holds 'not'( a 'eqv' b) = (a 'eqv' 'not' b) proof let a,b be Function of Y,BOOLEAN; let x be Element of Y; ('not'( a 'eqv' b)).x =('not'( (a 'imp' b) '&' (b 'imp' a))).x by BVFUNC_4:7 .=('not'( ('not' a 'or' b) '&' (b 'imp' a))).x by BVFUNC_4:8 .=('not'( ('not' a 'or' b) '&' ('not' b 'or' a))).x by BVFUNC_4:8 .=('not'('not' a 'or' b) 'or' 'not'('not' b 'or' a)).x by BVFUNC_1:14 .=(('not' 'not' a '&' 'not' b) 'or' 'not'('not' b 'or' a)).x by BVFUNC_1:13 .=((a '&' 'not' b) 'or' ('not' 'not' b '&' 'not' a)).x by BVFUNC_1:13 .=(((a '&' 'not' b) 'or' b) '&' ((a '&' 'not' b) 'or' 'not' a)).x by BVFUNC_1:11 .=(((a 'or' b) '&' ('not' b 'or' b)) '&' ((a '&' 'not' b) 'or' 'not' a )).x by BVFUNC_1:11 .=(((a 'or' b) '&' ('not' b 'or' b)) '&' ((a 'or' 'not' a) '&' ('not' b 'or' 'not' a))).x by BVFUNC_1:11 .=(((a 'or' b) '&' I_el(Y)) '&' ((a 'or' 'not' a) '&' ('not' b 'or' 'not' a))).x by BVFUNC_4:6 .=(((a 'or' b) '&' I_el(Y)) '&' (I_el(Y) '&' ('not' b 'or' 'not' a))). x by BVFUNC_4:6 .=((a 'or' b) '&' (I_el(Y) '&' ('not' b 'or' 'not' a))).x by BVFUNC_1:6 .=(('not' a 'or' 'not' b) '&' ('not' 'not' b 'or' a)).x by BVFUNC_1:6 .=(('not' a 'or' 'not' b) '&' ('not' b 'imp' a)).x by BVFUNC_4:8 .=((a 'imp' 'not' b) '&' ('not' b 'imp' a)).x by BVFUNC_4:8 .=(a 'eqv' 'not' b).x by BVFUNC_4:7; hence thesis; end; theorem for a,b being Function of Y,BOOLEAN holds 'not' a '<' (a 'imp' b ) 'eqv' 'not' a proof let a,b be Function of Y,BOOLEAN; let z be Element of Y; assume A1: ('not' a).z=TRUE; ((a 'imp' b) 'eqv' 'not' a).z =(('not' a 'or' b) 'eqv' 'not' a).z by BVFUNC_4:8 .=((('not' a 'or' b) 'imp' 'not' a) '&' ('not' a 'imp' ('not' a 'or' b)) ).z by BVFUNC_4:7 .=(('not'('not' a 'or' b) 'or' 'not' a) '&' ('not' a 'imp' ('not' a 'or' b))).z by BVFUNC_4:8 .=(('not'('not' a 'or' b) 'or' 'not' a) '&' ('not' 'not' a 'or' ('not' a 'or' b))).z by BVFUNC_4:8 .=('not'('not' a 'or' b) 'or' 'not' a).z '&' ('not' 'not' a 'or' ('not' a 'or' b)).z by MARGREL1:def 20 .=(('not'('not' a 'or' b)).z 'or' ('not' a).z) '&' ('not' 'not' a 'or' ( 'not' a 'or' b)).z by BVFUNC_1:def 4 .=('not' ('not' a 'or' b).z 'or' ('not' a).z) '&' ('not' 'not' a 'or' ( 'not' a 'or' b)).z by MARGREL1:def 19 .=('not'( ('not' a).z 'or' b.z) 'or' ('not' a).z) '&' ('not' 'not' a 'or' ('not' a 'or' b)).z by BVFUNC_1:def 4 .=(('not' 'not' a.z '&' 'not' b.z) 'or' ('not' a).z) '&' ('not' 'not' a 'or' ('not' a 'or' b)).z by MARGREL1:def 19 .=((a.z '&' 'not' b.z) 'or' ('not' a).z) '&' (('not' 'not' a).z 'or' ('not' a 'or' b).z) by BVFUNC_1:def 4 .=((a.z '&' 'not' b.z) 'or' ('not' a).z) '&' (a.z 'or' (('not' a). z 'or' b.z)) by BVFUNC_1:def 4 .=TRUE '&' (FALSE 'or' (TRUE 'or' b.z)) by A1 .=FALSE 'or' (TRUE 'or' b.z) .=(TRUE 'or' b.z) .=TRUE; hence thesis; end; theorem for a,b being Function of Y,BOOLEAN holds 'not' a '<' (b 'imp' a ) 'eqv' 'not' b proof let a,b be Function of Y,BOOLEAN; let z be Element of Y; assume ('not' a).z=TRUE; then A1: 'not' a.z=TRUE by MARGREL1:def 19; ((b 'imp' a) 'eqv' 'not' b).z =(('not' b 'or' a) 'eqv' 'not' b).z by BVFUNC_4:8 .=((('not' b 'or' a) 'imp' 'not' b) '&' ('not' b 'imp' ('not' b 'or' a)) ).z by BVFUNC_4:7 .=(('not'('not' b 'or' a) 'or' 'not' b) '&' ('not' b 'imp' ('not' b 'or' a))).z by BVFUNC_4:8 .=(('not'('not' b 'or' a) 'or' 'not' b) '&' ('not' 'not' b 'or' ('not' b 'or' a))).z by BVFUNC_4:8 .=('not'('not' b 'or' a) 'or' 'not' b).z '&' ('not' 'not' b 'or' ('not' b 'or' a)).z by MARGREL1:def 20 .=(('not'('not' b 'or' a)).z 'or' ('not' b).z) '&' ('not' 'not' b 'or' ( 'not' b 'or' a)).z by BVFUNC_1:def 4 .=('not' ('not' b 'or' a).z 'or' ('not' b).z) '&' ('not' 'not' b 'or' ( 'not' b 'or' a)).z by MARGREL1:def 19 .=('not'( ('not' b).z 'or' a.z) 'or' ('not' b).z) '&' ('not' 'not' b 'or' ('not' b 'or' a)).z by BVFUNC_1:def 4 .=(('not' 'not' b.z '&' 'not' a.z) 'or' ('not' b).z) '&' ('not' 'not' b 'or' ('not' b 'or' a)).z by MARGREL1:def 19 .=((b.z '&' 'not' a.z) 'or' ('not' b).z) '&' (('not' 'not' b).z 'or' ('not' b 'or' a).z) by BVFUNC_1:def 4 .=((b.z '&' 'not' a.z) 'or' ('not' b).z) '&' (('not' 'not' b).z 'or' (('not' b).z 'or' a.z)) by BVFUNC_1:def 4 .=((b.z '&' 'not' a.z) 'or' ('not' b).z) '&' (b.z 'or' ('not' b. z 'or' a.z)) by MARGREL1:def 19 .=((TRUE '&' b.z) 'or' ('not' b).z) '&' (b.z 'or' ('not' b.z 'or' FALSE)) by A1 .=(b.z 'or' ('not' b).z) '&' (b.z 'or' ('not' b.z 'or' FALSE)) .=(b.z 'or' 'not' b.z) '&' (b.z 'or' ('not' b.z 'or' FALSE)) by MARGREL1:def 19 .=TRUE '&' (b.z 'or' ('not' b.z 'or' FALSE)) .=b.z 'or' ('not' b.z 'or' FALSE) .=b.z 'or' 'not' b.z 'or' FALSE .=TRUE 'or' FALSE by XBOOLEAN:102 .=TRUE; hence thesis; end; theorem for a,b being Function of Y,BOOLEAN holds a '<' (a 'or' b) 'eqv' (b 'or' a) 'eqv' a proof let a,b be Function of Y,BOOLEAN; let z be Element of Y; assume A1: a.z=TRUE; A2: ((a 'or' b) 'eqv' (b 'or' a)).z =(((a 'or' b) 'imp' (a 'or' b)) '&' ((a 'or' b) 'imp' (a 'or' b))).z by BVFUNC_4:7 .=('not'( a 'or' b) 'or' (a 'or' b)).z by BVFUNC_4:8 .=(I_el(Y)).z by BVFUNC_4:6 .=TRUE by BVFUNC_1:def 11; ((a 'or' b) 'eqv' (b 'or' a) 'eqv' a).z =((((a 'or' b) 'eqv' (a 'or' b)) 'imp' a) '&' (a 'imp' ((a 'or' b) 'eqv' (a 'or' b)))).z by BVFUNC_4:7 .=(((a 'or' b) 'eqv' (a 'or' b)) 'imp' a).z '&' (a 'imp' ((a 'or' b) 'eqv' (a 'or' b))).z by MARGREL1:def 20 .=('not'( (a 'or' b) 'eqv' (a 'or' b)) 'or' a).z '&' (a 'imp' ((a 'or' b ) 'eqv' (a 'or' b))).z by BVFUNC_4:8 .=('not'( (a 'or' b) 'eqv' (a 'or' b)) 'or' a).z '&' ('not' a 'or' ((a 'or' b) 'eqv' (a 'or' b))).z by BVFUNC_4:8 .=(('not'( (a 'or' b) 'eqv' (a 'or' b))).z 'or' a.z) '&' ('not' a 'or' ((a 'or' b) 'eqv' (a 'or' b))).z by BVFUNC_1:def 4 .=(('not'( (a 'or' b) 'eqv' (a 'or' b))).z 'or' a.z) '&' (('not' a).z 'or' ((a 'or' b) 'eqv' (a 'or' b)).z) by BVFUNC_1:def 4 .=('not' ((a 'or' b) 'eqv' (a 'or' b)).z 'or' a.z) '&' (('not' a).z 'or' ((a 'or' b) 'eqv' (a 'or' b)).z) by MARGREL1:def 19 .=(FALSE 'or' a.z) '&' (('not' a).z 'or' TRUE) by A2 .=a.z '&' (('not' a).z 'or' TRUE) .=a.z '&' TRUE .=TRUE by A1; hence thesis; end; theorem for a being Function of Y,BOOLEAN holds a 'imp' ('not' a 'eqv' 'not' a) = I_el(Y) proof let a be Function of Y,BOOLEAN; for x being Element of Y holds (a 'imp' ('not' a 'eqv' 'not' a)).x = TRUE proof let x be Element of Y; (a 'imp' ('not' a 'eqv' 'not' a)).x =('not' a 'or' ('not' a 'eqv' 'not' a)).x by BVFUNC_4:8 .=('not' a 'or' (('not' a 'imp' 'not' a) '&' ('not' a 'imp' 'not' a))) .x by BVFUNC_4:7 .=('not' a 'or' ('not' 'not' a 'or' 'not' a)).x by BVFUNC_4:8 .=('not' a 'or' I_el(Y)).x by BVFUNC_4:6 .=TRUE by BVFUNC_1:10,def 11; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b being Function of Y,BOOLEAN holds ((a 'imp' b) 'imp' a) 'imp' a = I_el(Y) proof let a,b be Function of Y,BOOLEAN; for x being Element of Y holds (((a 'imp' b) 'imp' a) 'imp' a).x = TRUE proof let x be Element of Y; (((a 'imp' b) 'imp' a) 'imp' a).x =('not'( (a 'imp' b) 'imp' a) 'or' a ).x by BVFUNC_4:8 .=('not'('not'( a 'imp' b) 'or' a) 'or' a).x by BVFUNC_4:8 .=('not'('not'('not' a 'or' b) 'or' a) 'or' a).x by BVFUNC_4:8 .=('not'( ('not' 'not' a '&' 'not' b) 'or' a) 'or' a).x by BVFUNC_1:13 .=('not'( ((a 'or' a) '&' ('not' b 'or' a))) 'or' a).x by BVFUNC_1:11 .=(('not' a 'or' 'not'('not' b 'or' a)) 'or' a).x by BVFUNC_1:14 .=(('not' a 'or' ('not' 'not' b '&' 'not' a)) 'or' a).x by BVFUNC_1:13 .=((('not' a 'or' b) '&' ('not' a 'or' 'not' a)) 'or' a).x by BVFUNC_1:11 .=((('not' a 'or' b) 'or' a) '&' ('not' a 'or' a)).x by BVFUNC_1:11 .=((('not' a 'or' b) 'or' a) '&' I_el(Y)).x by BVFUNC_4:6 .=(('not' a 'or' b) 'or' a).x by BVFUNC_1:6 .=(b 'or' ('not' a 'or' a)).x by BVFUNC_1:8 .=(b 'or' I_el(Y)).x by BVFUNC_4:6 .=TRUE by BVFUNC_1:10,def 11; hence thesis; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b,c,d being Function of Y,BOOLEAN holds ((a 'imp' c) '&' ( b 'imp' d)) '&' ('not' c 'or' 'not' d) 'imp' ('not' a 'or' 'not' b)=I_el(Y) proof let a,b,c,d be Function of Y,BOOLEAN; for x being Element of Y holds (((a 'imp' c) '&' (b 'imp' d)) '&' ('not' c 'or' 'not' d) 'imp' ('not' a 'or' 'not' b)).x = TRUE proof let x be Element of Y; ((a 'imp' c) '&' (b 'imp' d)) '&' ('not' c 'or' 'not' d) 'imp' ('not' a 'or' 'not' b) ='not'( ((a 'imp' c) '&' (b 'imp' d)) '&' ('not' c 'or' 'not' d )) 'or' ('not' a 'or' 'not' b) by BVFUNC_4:8 .='not'( (('not' a 'or' c) '&' (b 'imp' d)) '&' ('not' c 'or' 'not' d) ) 'or' ('not' a 'or' 'not' b) by BVFUNC_4:8 .='not'( (('not' a 'or' c) '&' ('not' b 'or' d)) '&' ('not' c 'or' 'not' d)) 'or' ('not' a 'or' 'not' b) by BVFUNC_4:8 .=('not'( ('not' a 'or' c) '&' ('not' b 'or' d)) 'or' 'not'('not' c 'or' 'not' d)) 'or' ('not' a 'or' 'not' b) by BVFUNC_1:14 .=(('not'('not' a 'or' c) 'or' 'not'('not' b 'or' d)) 'or' 'not'('not' c 'or' 'not' d)) 'or' ('not' a 'or' 'not' b) by BVFUNC_1:14 .=((('not' 'not' a '&' 'not' c) 'or' 'not'('not' b 'or' d)) 'or' 'not' ('not' c 'or' 'not' d)) 'or' ('not' a 'or' 'not' b) by BVFUNC_1:13 .=(((a '&' 'not' c) 'or' ('not' 'not' b '&' 'not' d)) 'or' 'not'('not' c 'or' 'not' d)) 'or' ('not' a 'or' 'not' b) by BVFUNC_1:13 .=(((a '&' 'not' c) 'or' (b '&' 'not' d)) 'or' ('not' 'not' c '&' 'not' 'not' d)) 'or' ('not' a 'or' 'not' b) by BVFUNC_1:13 .=((a '&' 'not' c) 'or' ((b '&' 'not' d) 'or' (c '&' d))) 'or' ('not' a 'or' 'not' b) by BVFUNC_1:8 .=((a '&' 'not' c) 'or' ((b 'or' (c '&' d)) '&' ('not' d 'or' (c '&' d )))) 'or' ('not' a 'or' 'not' b) by BVFUNC_1:11 .=((a '&' 'not' c) 'or' ((b 'or' (c '&' d)) '&' (('not' d 'or' c) '&' ('not' d 'or' d)))) 'or' ('not' a 'or' 'not' b) by BVFUNC_1:11 .=((a '&' 'not' c) 'or' ((b 'or' (c '&' d)) '&' (('not' d 'or' c) '&' I_el(Y)))) 'or' ('not' a 'or' 'not' b) by BVFUNC_4:6 .=((a '&' 'not' c) 'or' ((b 'or' (c '&' d)) '&' ('not' d 'or' c))) 'or' ('not' a 'or' 'not' b) by BVFUNC_1:6 .=((b 'or' (c '&' d)) '&' ('not' d 'or' c)) 'or' ((a '&' 'not' c) 'or' ('not' a 'or' 'not' b)) by BVFUNC_1:8 .=((b 'or' (c '&' d)) '&' ('not' d 'or' c)) 'or' ((a 'or' ('not' a 'or' 'not' b)) '&' ('not' c 'or' ('not' a 'or' 'not' b))) by BVFUNC_1:11 .=((b 'or' (c '&' d)) '&' ('not' d 'or' c)) 'or' (((a 'or' 'not' a) 'or' 'not' b) '&' ('not' c 'or' ('not' a 'or' 'not' b))) by BVFUNC_1:8 .=((b 'or' (c '&' d)) '&' ('not' d 'or' c)) 'or' ((I_el(Y) 'or' 'not' b) '&' ('not' c 'or' ('not' a 'or' 'not' b))) by BVFUNC_4:6 .=((b 'or' (c '&' d)) '&' ('not' d 'or' c)) 'or' (I_el(Y) '&' ('not' c 'or' ('not' a 'or' 'not' b))) by BVFUNC_1:10 .=((b 'or' (c '&' d)) '&' ('not' d 'or' c)) 'or' ('not' c 'or' ('not' a 'or' 'not' b)) by BVFUNC_1:6 .=((b 'or' (c '&' d)) 'or' ('not' c 'or' ('not' a 'or' 'not' b))) '&' (('not' d 'or' c) 'or' ('not' c 'or' ('not' a 'or' 'not' b))) by BVFUNC_1:11 .=((b 'or' (c '&' d)) 'or' ('not' c 'or' ('not' a 'or' 'not' b))) '&' ((('not' d 'or' c) 'or' 'not' c) 'or' ('not' a 'or' 'not' b)) by BVFUNC_1:8 .=((b 'or' (c '&' d)) 'or' ('not' c 'or' ('not' a 'or' 'not' b))) '&' (('not' d 'or' (c 'or' 'not' c)) 'or' ('not' a 'or' 'not' b)) by BVFUNC_1:8 .=((b 'or' (c '&' d)) 'or' ('not' c 'or' ('not' a 'or' 'not' b))) '&' (('not' d 'or' I_el(Y)) 'or' ('not' a 'or' 'not' b)) by BVFUNC_4:6 .=((b 'or' (c '&' d)) 'or' ('not' c 'or' ('not' a 'or' 'not' b))) '&' (I_el(Y) 'or' ('not' a 'or' 'not' b)) by BVFUNC_1:10 .=((b 'or' (c '&' d)) 'or' ('not' c 'or' ('not' a 'or' 'not' b))) '&' I_el(Y) by BVFUNC_1:10 .=(b 'or' (c '&' d)) 'or' ('not' c 'or' ('not' a 'or' 'not' b)) by BVFUNC_1:6 .=(c '&' d) 'or' (b 'or' ('not' c 'or' ('not' a 'or' 'not' b))) by BVFUNC_1:8 .=(c '&' d) 'or' ((b 'or' ('not' b 'or' 'not' a)) 'or' 'not' c) by BVFUNC_1:8 .=(c '&' d) 'or' (((b 'or' 'not' b) 'or' 'not' a) 'or' 'not' c) by BVFUNC_1:8 .=(c '&' d) 'or' ((I_el(Y) 'or' 'not' a) 'or' 'not' c) by BVFUNC_4:6 .=(c '&' d) 'or' (I_el(Y) 'or' 'not' c) by BVFUNC_1:10 .=(c '&' d) 'or' I_el(Y) by BVFUNC_1:10 .=I_el(Y) by BVFUNC_1:10; hence thesis by BVFUNC_1:def 11; end; hence thesis by BVFUNC_1:def 11; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) 'imp' (( a 'imp' (b 'imp' c)) 'imp' (a 'imp' c)) = I_el(Y) proof let a,b,c be Function of Y,BOOLEAN; for x being Element of Y holds ((a 'imp' b) 'imp' ((a 'imp' (b 'imp' c)) 'imp' (a 'imp' c))).x=TRUE proof let x be Element of Y; (a 'imp' b) 'imp' ((a 'imp' (b 'imp' c)) 'imp' (a 'imp' c)) ='not' (a 'imp' b) 'or' ((a 'imp' (b 'imp' c)) 'imp' (a 'imp' c)) by BVFUNC_4:8 .='not'('not' a 'or' b) 'or' ((a 'imp' (b 'imp' c)) 'imp' (a 'imp' c)) by BVFUNC_4:8 .='not'('not' a 'or' b) 'or' (('not' a 'or' (b 'imp' c)) 'imp' (a 'imp' c)) by BVFUNC_4:8 .='not'('not' a 'or' b) 'or' (('not' a 'or' ('not' b 'or' c)) 'imp' (a 'imp' c)) by BVFUNC_4:8 .='not'('not' a 'or' b) 'or' (('not' a 'or' ('not' b 'or' c)) 'imp' ( 'not' a 'or' c)) by BVFUNC_4:8 .='not'('not' a 'or' b) 'or' ('not'('not' a 'or' ('not' b 'or' c)) 'or' ('not' a 'or' c)) by BVFUNC_4:8 .=('not' 'not' a '&' 'not' b) 'or' ('not'('not' a 'or' ('not' b 'or' c )) 'or' ( 'not' a 'or' c)) by BVFUNC_1:13 .=('not' 'not' a '&' 'not' b) 'or' (('not' 'not' a '&' 'not'('not' b 'or' c)) 'or' ('not' a 'or' c)) by BVFUNC_1:13 .=(a '&' 'not' b) 'or' (('not' 'not' a '&' ('not' 'not' b '&' 'not' c) ) 'or' ('not' a 'or' c)) by BVFUNC_1:13 .=(a '&' 'not' b) 'or' ((a 'or' ('not' a 'or' c)) '&' ((b '&' 'not' c) 'or' ('not' a 'or' c))) by BVFUNC_1:11 .=(a '&' 'not' b) 'or' (((a 'or' 'not' a) 'or' c) '&' ((b '&' 'not' c) 'or' ('not' a 'or' c))) by BVFUNC_1:8 .=(a '&' 'not' b) 'or' ((I_el(Y) 'or' c) '&' ((b '&' 'not' c) 'or' ( 'not' a 'or' c))) by BVFUNC_4:6 .=(a '&' 'not' b) 'or' (I_el(Y) '&' ((b '&' 'not' c) 'or' ('not' a 'or' c))) by BVFUNC_1:10 .=(a '&' 'not' b) 'or' ((b '&' 'not' c) 'or' ('not' a 'or' c)) by BVFUNC_1:6 .=(a '&' 'not' b) 'or' ((b 'or' ('not' a 'or' c)) '&' ('not' c 'or' ( 'not' a 'or' c))) by BVFUNC_1:11 .=(a '&' 'not' b) 'or' ((b 'or' ('not' a 'or' c)) '&' (('not' c 'or' c ) 'or' 'not' a)) by BVFUNC_1:8 .=(a '&' 'not' b) 'or' ((b 'or' ('not' a 'or' c)) '&' (I_el(Y) 'or' 'not' a)) by BVFUNC_4:6 .=(a '&' 'not' b) 'or' ((b 'or' ('not' a 'or' c)) '&' I_el(Y)) by BVFUNC_1:10 .=(a '&' 'not' b) 'or' (b 'or' ('not' a 'or' c)) by BVFUNC_1:6 .=(a 'or' (b 'or' ('not' a 'or' c))) '&' ('not' b 'or' (b 'or' ('not' a 'or' c))) by BVFUNC_1:11 .=(a 'or' (b 'or' ('not' a 'or' c))) '&' (('not' b 'or' b) 'or' ('not' a 'or' c)) by BVFUNC_1:8 .=(a 'or' (b 'or' ('not' a 'or' c))) '&' (I_el(Y) 'or' ('not' a 'or' c )) by BVFUNC_4:6 .=(a 'or' (b 'or' ('not' a 'or' c))) '&' I_el(Y) by BVFUNC_1:10 .=a 'or' (b 'or' ('not' a 'or' c)) by BVFUNC_1:6 .=a 'or' (('not' a 'or' b) 'or' c) by BVFUNC_1:8 .=a 'or' ('not' a 'or' b) 'or' c by BVFUNC_1:8 .=(a 'or' 'not' a) 'or' b 'or' c by BVFUNC_1:8 .=I_el(Y) 'or' b 'or' c by BVFUNC_4:6 .=I_el(Y) 'or' c by BVFUNC_1:10 .=I_el(Y) by BVFUNC_1:10; hence thesis by BVFUNC_1:def 11; end; hence thesis by BVFUNC_1:def 11; end; begin :: BVFUNC_9 reserve Y for non empty set, a,b,c,d,e,f,g for Function of Y,BOOLEAN; Lm1: a '&' b '<' a proof let x be Element of Y; assume (a '&' b).x = TRUE; then a.x '&' b.x = TRUE by MARGREL1:def 20; hence thesis by MARGREL1:12; end; Lm2: a '&' b '&' c '<' a & a '&' b '&' c '<' b proof a '&' b '&' c = c '&' b '&' a & c '&' b '&' a 'imp' a = I_el(Y) by BVFUNC_1:4 ,Th38; hence a '&' b '&' c '<' a by BVFUNC_1:16; a '&' b '&' c = a '&' c '&' b & a '&' c '&' b 'imp' b = I_el(Y) by BVFUNC_1:4 ,Th38; hence thesis by BVFUNC_1:16; end; Lm3: a '&' b '&' c '&' d '<' a & a '&' b '&' c '&' d '<' b proof A1: d '&' c '&' b '&' a 'imp' a = I_el(Y) by Th38; a '&' b '&' c '&' d = d '&' c '&' (b '&' a) by BVFUNC_1:4 .=d '&' c '&' b '&' a by BVFUNC_1:4; hence a '&' b '&' c '&' d '<' a by A1,BVFUNC_1:16; A2: a '&' c '&' d '&' b 'imp' b = I_el(Y) by Th38; a '&' b '&' c '&' d = a '&' c '&' b '&' d by BVFUNC_1:4 .=a '&' c '&' d '&' b by BVFUNC_1:4; hence thesis by A2,BVFUNC_1:16; end; Lm4: a '&' b '&' c '&' d '&' e '<' a & a '&' b '&' c '&' d '&' e '<' b proof A1: e '&' d '&' c '&' b '&' a 'imp' a = I_el(Y) by Th38; a '&' b '&' c '&' d '&' e =e '&' d '&' (c '&' (b '&' a)) by BVFUNC_1:4 .=e '&' d '&' c '&' (b '&' a) by BVFUNC_1:4 .=e '&' d '&' c '&' b '&' a by BVFUNC_1:4; hence a '&' b '&' c '&' d '&' e '<' a by A1,BVFUNC_1:16; A2: a '&' c '&' d '&' e '&' b 'imp' b = I_el(Y) by Th38; a '&' b '&' c '&' d '&' e =a '&' c '&' b '&' d '&' e by BVFUNC_1:4 .=a '&' c '&' d '&' b '&' e by BVFUNC_1:4 .=a '&' c '&' d '&' e '&' b by BVFUNC_1:4; hence thesis by A2,BVFUNC_1:16; end; Lm5: a '&' b '&' c '&' d '&' e '&' f '<' a & a '&' b '&' c '&' d '&' e '&' f '<' b proof A1: f '&' e '&' d '&' c '&' b '&' a 'imp' a = I_el(Y) by Th38; a '&' b '&' c '&' d '&' e '&' f =f '&' e '&' (d '&' (c '&' (b '&' a))) by BVFUNC_1:4 .=f '&' e '&' d '&' (c '&' (b '&' a)) by BVFUNC_1:4 .=f '&' e '&' d '&' c '&' (b '&' a) by BVFUNC_1:4 .=f '&' e '&' d '&' c '&' b '&' a by BVFUNC_1:4; hence a '&' b '&' c '&' d '&' e '&' f '<' a by A1,BVFUNC_1:16; A2: f '&' e '&' d '&' c '&' a '&' b 'imp' b = I_el(Y) by Th38; a '&' b '&' c '&' d '&' e '&' f =f '&' e '&' (d '&' (c '&' (b '&' a))) by BVFUNC_1:4 .=f '&' e '&' d '&' (c '&' (b '&' a)) by BVFUNC_1:4 .=f '&' e '&' d '&' c '&' (b '&' a) by BVFUNC_1:4 .=f '&' e '&' d '&' c '&' a '&' b by BVFUNC_1:4; hence thesis by A2,BVFUNC_1:16; end; Lm6: a '&' b '&' c '&' d '&' e '&' f '&' g '<' a & a '&' b '&' c '&' d '&' e '&' f '&' g '<' b proof A1: g '&' f '&' e '&' d '&' c '&' b '&' a 'imp' a = I_el(Y) by Th38; a '&' b '&' c '&' d '&' e '&' f '&' g =g '&' f '&' (e '&' (d '&' (c '&' (b '&' a)))) by BVFUNC_1:4 .=g '&' f '&' e '&' (d '&' (c '&' (b '&' a))) by BVFUNC_1:4 .=g '&' f '&' e '&' d '&' (c '&' (b '&' a)) by BVFUNC_1:4 .=g '&' f '&' e '&' d '&' c '&' (b '&' a) by BVFUNC_1:4 .=g '&' f '&' e '&' d '&' c '&' b '&' a by BVFUNC_1:4; hence a '&' b '&' c '&' d '&' e '&' f '&' g '<' a by A1,BVFUNC_1:16; A2: a '&' g '&' f '&' e '&' d '&' c '&' b 'imp' b = I_el(Y) by Th38; a '&' b '&' c '&' d '&' e '&' f '&' g =a '&' (c '&' b) '&' d '&' e '&' f '&' g by BVFUNC_1:4 .=a '&' (d '&' (c '&' b)) '&' e '&' f '&' g by BVFUNC_1:4 .=a '&' (e '&' (d '&' (c '&' b))) '&' f '&' g by BVFUNC_1:4 .=a '&' (f '&' (e '&' (d '&' (c '&' b)))) '&' g by BVFUNC_1:4 .=a '&' g '&' (f '&' (e '&' (d '&' (c '&' b)))) by BVFUNC_1:4 .=a '&' g '&' f '&' (e '&' (d '&' (c '&' b))) by BVFUNC_1:4 .=a '&' g '&' f '&' e '&' (d '&' (c '&' b)) by BVFUNC_1:4 .=a '&' g '&' f '&' e '&' d '&' (c '&' b) by BVFUNC_1:4 .=a '&' g '&' f '&' e '&' d '&' c '&' b by BVFUNC_1:4; hence thesis by A2,BVFUNC_1:16; end; theorem Th1: (a 'or' b) '&' (b 'imp' c) '<' a 'or' c proof let z be Element of Y; A1: ((a 'or' b) '&' (b 'imp' c)).z =(a 'or' b).z '&' (b 'imp' c).z by MARGREL1:def 20 .=(a 'or' b).z '&' ('not' b 'or' c).z by BVFUNC_4:8 .=(a 'or' b).z '&' (('not' b).z 'or' c.z) by BVFUNC_1:def 4 .=(a.z 'or' b.z) '&' (('not' b).z 'or' c.z) by BVFUNC_1:def 4; assume A2: ((a 'or' b) '&' (b 'imp' c)).z=TRUE; now assume (a 'or' c).z<>TRUE; then (a 'or' c).z=FALSE by XBOOLEAN:def 3; then A3: a.z 'or' c.z=FALSE by BVFUNC_1:def 4; c.z=TRUE or c.z=FALSE by XBOOLEAN:def 3; then (a.z 'or' b.z) '&' (('not' b).z 'or' c.z) =b.z '&' 'not' b.z by A3, MARGREL1:def 19 .=FALSE by XBOOLEAN:138; hence contradiction by A2,A1; end; hence thesis; end; theorem Th2: a '&' (a 'imp' b) '<' b proof let z be Element of Y; A1: (a '&' (a 'imp' b)).z =a.z '&' (a 'imp' b).z by MARGREL1:def 20 .=a.z '&' ('not' a 'or' b).z by BVFUNC_4:8 .=a.z '&' (('not' a).z 'or' b.z) by BVFUNC_1:def 4 .=a.z '&' ('not' a).z 'or' a.z '&' b.z by XBOOLEAN:8 .=a.z '&' 'not' a.z 'or' a.z '&' b.z by MARGREL1:def 19 .=FALSE 'or' a.z '&' b.z by XBOOLEAN:138 .=a.z '&' b.z; assume A2: (a '&' (a 'imp' b)).z=TRUE; now assume b.z<>TRUE; then a.z '&' b.z = FALSE '&' a.z by XBOOLEAN:def 3 .= FALSE; hence contradiction by A2,A1; end; hence thesis; end; theorem (a 'imp' b) '&' 'not' b '<' 'not' a proof let z be Element of Y; reconsider bz = b.z as boolean object; A1: ((a 'imp' b) '&' 'not' b).z =(a 'imp' b).z '&' ('not' b).z by MARGREL1:def 20 .=('not' a 'or' b).z '&' ('not' b).z by BVFUNC_4:8 .=('not' b).z '&' (('not' a).z 'or' b.z) by BVFUNC_1:def 4 .=('not' b).z '&' ('not' a).z 'or' ('not' b).z '&' b.z by XBOOLEAN:8 .=('not' b).z '&' ('not' a).z 'or' 'not' bz '&' bz by MARGREL1:def 19 .=('not' b).z '&' ('not' a).z 'or' FALSE by XBOOLEAN:138 .=('not' b).z '&' ('not' a).z; assume A2: ((a 'imp' b) '&' 'not' b).z=TRUE; now assume ('not' a).z<>TRUE; then ('not' b).z '&' ('not' a).z =FALSE '&' ('not' b).z by XBOOLEAN:def 3 .=FALSE; hence contradiction by A2,A1; end; hence thesis; end; theorem (a 'or' b) '&' 'not' a '<' b proof let z be Element of Y; reconsider az = a.z as boolean object; A1: ((a 'or' b) '&' 'not' a).z =(a 'or' b).z '&' ('not' a).z by MARGREL1:def 20 .=('not' a).z '&' (a.z 'or' b.z) by BVFUNC_1:def 4 .=('not' a).z '&' a.z 'or' ('not' a).z '&' b.z by XBOOLEAN:8 .='not' az '&' az 'or' ('not' a).z '&' b.z by MARGREL1:def 19 .=FALSE 'or' ('not' a).z '&' b.z by XBOOLEAN:138 .=('not' a).z '&' b.z; assume A2: ((a 'or' b) '&' 'not' a).z=TRUE; now assume b.z<>TRUE; then ('not' a).z '&' b.z =FALSE '&' ('not' a).z by XBOOLEAN:def 3 .=FALSE; hence contradiction by A2,A1; end; hence thesis; end; theorem (a 'imp' b) '&' ('not' a 'imp' b) '<' b proof let z be Element of Y; reconsider az = a.z as boolean object; A1: ((a 'imp' b) '&' ('not' a 'imp' b)).z =(a 'imp' b).z '&' ('not' a 'imp' b).z by MARGREL1:def 20 .=('not' a 'or' b).z '&' ('not' a 'imp' b).z by BVFUNC_4:8 .=('not' a 'or' b).z '&' ('not' 'not' a 'or' b).z by BVFUNC_4:8 .=(('not' a).z 'or' b.z) '&' (a 'or' b).z by BVFUNC_1:def 4 .=(('not' a).z 'or' b.z) '&' (a.z 'or' b.z) by BVFUNC_1:def 4; assume A2: ((a 'imp' b) '&' ('not' a 'imp' b)).z=TRUE; now assume b.z<>TRUE; then b.z=FALSE by XBOOLEAN:def 3; then (('not' a).z 'or' b.z) '&' (a.z 'or' b.z) ='not' az '&' az by MARGREL1:def 19 .=FALSE by XBOOLEAN:138; hence contradiction by A2,A1; end; hence thesis; end; theorem (a 'imp' b) '&' (a 'imp' 'not' b) '<' 'not' a proof let z be Element of Y; A1: ((a 'imp' b) '&' (a 'imp' 'not' b)).z =(a 'imp' b).z '&' (a 'imp' 'not' b).z by MARGREL1:def 20 .=('not' a 'or' b).z '&' (a 'imp' 'not' b).z by BVFUNC_4:8 .=('not' a 'or' b).z '&' ('not' a 'or' 'not' b).z by BVFUNC_4:8 .=(('not' a).z 'or' b.z) '&' ('not' a 'or' 'not' b).z by BVFUNC_1:def 4 .=(('not' a).z 'or' b.z) '&' (('not' a).z 'or' ('not' b).z) by BVFUNC_1:def 4; assume A2: ((a 'imp' b) '&' (a 'imp' 'not' b)).z=TRUE; now assume ('not' a).z<>TRUE; then ('not' a).z=FALSE by XBOOLEAN:def 3; then (('not' a).z 'or' b.z) '&' (('not' a).z 'or' ('not' b).z) =b.z '&' 'not' b.z by MARGREL1:def 19 .=FALSE by XBOOLEAN:138; hence contradiction by A2,A1; end; hence thesis; end; theorem a 'imp' (b '&' c) '<' a 'imp' b proof let z be Element of Y; A1: (a 'imp' (b '&' c)).z =('not' a 'or' (b '&' c)).z by BVFUNC_4:8 .=('not' a).z 'or' (b '&' c).z by BVFUNC_1:def 4 .=('not' a).z 'or' (b.z '&' c.z) by MARGREL1:def 20; assume A2: (a 'imp' (b '&' c)).z=TRUE; now assume (a 'imp' b).z<>TRUE; then (a 'imp' b).z=FALSE by XBOOLEAN:def 3; then ('not' a 'or' b).z=FALSE by BVFUNC_4:8; then A3: ('not' a).z 'or' b.z=FALSE by BVFUNC_1:def 4; ('not' a).z 'or' (b.z '&' c.z) =(('not' a).z 'or' b.z) '&' (('not' a). z 'or' c.z) by XBOOLEAN:9 .=FALSE by A3; hence contradiction by A2,A1; end; hence thesis; end; theorem (a 'or' b) 'imp' c '<' a 'imp' c proof let z be Element of Y; A1: ((a 'or' b) 'imp' c).z =('not'( a 'or' b) 'or' c).z by BVFUNC_4:8 .=(('not' a '&' 'not' b) 'or' c).z by BVFUNC_1:13 .=('not' a '&' 'not' b).z 'or' c.z by BVFUNC_1:def 4 .=(('not' a).z '&' ('not' b).z) 'or' c.z by MARGREL1:def 20; assume A2: ((a 'or' b) 'imp' c).z=TRUE; now assume (a 'imp' c).z<>TRUE; then (a 'imp' c).z=FALSE by XBOOLEAN:def 3; then ('not' a 'or' c).z=FALSE by BVFUNC_4:8; then A3: ('not' a).z 'or' c.z=FALSE by BVFUNC_1:def 4; (('not' a).z '&' ('not' b).z) 'or' c.z =(c.z 'or' ('not' a).z) '&' (c. z 'or' ('not' b).z) by XBOOLEAN:9 .=FALSE by A3; hence contradiction by A2,A1; end; hence thesis; end; theorem a 'imp' b '<' (a '&' c) 'imp' b proof let z be Element of Y; A1: (a 'imp' b).z =('not' a 'or' b).z by BVFUNC_4:8 .=('not' a).z 'or' b.z by BVFUNC_1:def 4; assume A2: (a 'imp' b).z=TRUE; now assume ((a '&' c) 'imp' b).z<>TRUE; then ((a '&' c) 'imp' b).z=FALSE by XBOOLEAN:def 3; then ('not'( a '&' c) 'or' b).z=FALSE by BVFUNC_4:8; then ('not'( a '&' c)).z 'or' b.z=FALSE by BVFUNC_1:def 4; then ('not' a 'or' 'not' c) .z 'or' b.z=FALSE by BVFUNC_1:14; then (('not' c).z 'or' ('not' a).z) 'or' b.z=FALSE by BVFUNC_1:def 4; then ('not' c).z 'or' (('not' a).z 'or' b.z)=FALSE; hence contradiction by A2,A1; end; hence thesis; end; theorem a 'imp' b '<' (a '&' c) 'imp' (b '&' c) proof let z be Element of Y; A1: (a 'imp' b).z =('not' a 'or' b).z by BVFUNC_4:8 .=('not' a).z 'or' b.z by BVFUNC_1:def 4; assume A2: (a 'imp' b).z=TRUE; now assume ((a '&' c) 'imp' (b '&' c)).z<>TRUE; then ((a '&' c) 'imp' (b '&' c)).z=FALSE by XBOOLEAN:def 3; then ('not'( a '&' c) 'or' (b '&' c)).z=FALSE by BVFUNC_4:8; then ('not'( a '&' c)).z 'or' (b '&' c).z=FALSE by BVFUNC_1:def 4; then ('not' a 'or' 'not' c) .z 'or' (b '&' c).z=FALSE by BVFUNC_1:14; then (('not' c).z 'or' ('not' a).z) 'or' (b '&' c).z=FALSE by BVFUNC_1:def 4; then ('not' c).z 'or' (('not' a).z 'or' (b '&' c).z)=FALSE; then A3: ('not' c).z 'or' (('not' a).z 'or' (b.z '&' c.z))=FALSE by MARGREL1:def 20; ('not' c).z 'or' ((('not' a).z 'or' b.z) '&' (('not' a).z 'or' c.z)) =('not' c).z 'or' c.z 'or' ('not' a).z by A2,A1 .='not' c.z 'or' c.z 'or' ('not' a).z by MARGREL1:def 19 .=TRUE 'or' ('not' a).z by XBOOLEAN:102 .=TRUE; hence contradiction by A3,XBOOLEAN:9; end; hence thesis; end; theorem a 'imp' b '<' a 'imp' (b 'or' c) proof let z be Element of Y; A1: (a 'imp' b).z =('not' a 'or' b).z by BVFUNC_4:8 .=('not' a).z 'or' b.z by BVFUNC_1:def 4; assume A2: (a 'imp' b).z=TRUE; now assume A3: (a 'imp' (b 'or' c)).z<>TRUE; (a 'imp' (b 'or' c)).z =('not' a 'or' (b 'or' c)).z by BVFUNC_4:8 .=('not' a).z 'or' (b 'or' c).z by BVFUNC_1:def 4 .=('not' a).z 'or' (b.z 'or' c.z) by BVFUNC_1:def 4 .=(('not' a).z 'or' b.z) 'or' c.z .=TRUE by A2,A1; hence contradiction by A3; end; hence thesis; end; theorem a 'imp' b '<' (a 'or' c) 'imp' (b 'or' c) proof let z be Element of Y; A1: (a 'imp' b).z =('not' a 'or' b).z by BVFUNC_4:8 .=('not' a).z 'or' b.z by BVFUNC_1:def 4; assume A2: (a 'imp' b).z=TRUE; now assume A3: ((a 'or' c) 'imp' (b 'or' c)).z<>TRUE; ((a 'or' c) 'imp' (b 'or' c)).z =('not'( a 'or' c) 'or' (b 'or' c)).z by BVFUNC_4:8 .=('not'( a 'or' c)).z 'or' (b 'or' c).z by BVFUNC_1:def 4 .=('not'( a 'or' c)).z 'or' (b.z 'or' c.z) by BVFUNC_1:def 4 .=(('not'( a 'or' c)).z 'or' b.z) 'or' c.z .=((('not' a '&' 'not' c)).z 'or' b.z) 'or' c.z by BVFUNC_1:13 .=(b.z 'or' (('not' a).z '&' ('not' c).z)) 'or' c.z by MARGREL1:def 20 .=((('not' a).z 'or' b.z) '&' (b.z 'or' ('not' c).z)) 'or' c.z by XBOOLEAN:9 .=b.z 'or' (('not' c).z 'or' c.z) by A2,A1 .=b.z 'or' ('not' c.z 'or' c.z) by MARGREL1:def 19 .=b.z 'or' TRUE by XBOOLEAN:102 .=TRUE; hence contradiction by A3; end; hence thesis; end; theorem a '&' b 'or' c '<' a 'or' c proof let z be Element of Y; A1: (a '&' b 'or' c).z =(a '&' b).z 'or' c.z by BVFUNC_1:def 4 .=(a.z '&' b.z) 'or' c.z by MARGREL1:def 20; assume A2: (a '&' b 'or' c).z=TRUE; now assume (a 'or' c).z<>TRUE; then (a 'or' c).z=FALSE by XBOOLEAN:def 3; then A3: a.z 'or' c.z=FALSE by BVFUNC_1:def 4; (a.z '&' b.z) 'or' c.z =(c.z 'or' a.z) '&' (c.z 'or' b.z) by XBOOLEAN:9 .=FALSE by A3; hence contradiction by A2,A1; end; hence thesis; end; theorem (a '&' b) 'or' (c '&' d) '<' a 'or' c proof let z be Element of Y; A1: ((a '&' b) 'or' (c '&' d)).z =(a '&' b).z 'or' (c '&' d).z by BVFUNC_1:def 4 .=(a.z '&' b.z) 'or' (c '&' d).z by MARGREL1:def 20 .=(a.z '&' b.z) 'or' (c.z '&' (d).z) by MARGREL1:def 20; assume A2: ((a '&' b) 'or' (c '&' d)).z=TRUE; now assume (a 'or' c).z<>TRUE; then (a 'or' c).z=FALSE by XBOOLEAN:def 3; then A3: a.z 'or' c.z=FALSE by BVFUNC_1:def 4; (a.z '&' b.z) 'or' (c.z '&' (d).z) =(c.z 'or' (a.z '&' b.z)) '&' ((a.z '&' b.z) 'or' (d).z) by XBOOLEAN:9 .=((a.z 'or' c.z) '&' (c.z 'or' b.z)) '&' ((a.z '&' b.z) 'or' (d).z) by XBOOLEAN:9 .=FALSE by A3; hence contradiction by A2,A1; end; hence thesis; end; theorem (a 'imp' b) '&' ('not' a 'imp' c) '<' b 'or' c proof let z be Element of Y; A1: ((a 'imp' b) '&' ('not' a 'imp' c)).z =(('not' a 'or' b) '&' ('not' a 'imp' c)).z by BVFUNC_4:8 .=(('not' a 'or' b) '&' ('not' 'not' a 'or' c)).z by BVFUNC_4:8 .=('not' a 'or' b).z '&' (a 'or' c).z by MARGREL1:def 20 .=(('not' a).z 'or' b.z) '&' (a 'or' c).z by BVFUNC_1:def 4 .=(('not' a).z 'or' b.z) '&' (a.z 'or' c.z) by BVFUNC_1:def 4; assume A2: ((a 'imp' b) '&' ('not' a 'imp' c)).z=TRUE; now reconsider az = a.z as boolean object; assume (b 'or' c).z<>TRUE; then (b 'or' c).z=FALSE by XBOOLEAN:def 3; then A3: b.z 'or' c.z=FALSE by BVFUNC_1:def 4; c.z=TRUE or c.z=FALSE by XBOOLEAN:def 3; then (('not' a).z 'or' b.z) '&' (a.z 'or' c.z) ='not' az '&' az by A3, MARGREL1:def 19 .=FALSE by XBOOLEAN:138; hence contradiction by A2,A1; end; hence thesis; end; theorem (a 'imp' c) '&' (b 'imp' 'not' c) '<' 'not' a 'or' 'not' b proof let z be Element of Y; A1: ((a 'imp' c) '&' (b 'imp' 'not' c)).z =(a 'imp' c).z '&' (b 'imp' 'not' c).z by MARGREL1:def 20 .=('not' a 'or' c).z '&' (b 'imp' 'not' c).z by BVFUNC_4:8 .=('not' a 'or' c).z '&' ('not' b 'or' 'not' c).z by BVFUNC_4:8 .=(('not' a).z 'or' c.z) '&' ('not' b 'or' 'not' c).z by BVFUNC_1:def 4 .=(('not' a).z 'or' c.z) '&' (('not' b).z 'or' ('not' c).z) by BVFUNC_1:def 4; assume A2: ((a 'imp' c) '&' (b 'imp' 'not' c)).z=TRUE; now assume ('not' a 'or' 'not' b).z<>TRUE; then ('not' a 'or' 'not' b).z=FALSE by XBOOLEAN:def 3; then A3: ('not' a).z 'or' ('not' b).z=FALSE by BVFUNC_1:def 4; ('not' b).z=TRUE or ('not' b).z=FALSE by XBOOLEAN:def 3; then (('not' a).z 'or' c.z) '&' (('not' b).z 'or' ('not' c).z) =c.z '&' 'not' c.z by A3,MARGREL1:def 19 .=FALSE by XBOOLEAN:138; hence contradiction by A2,A1; end; hence thesis; end; theorem (a 'or' b) '&' ('not' a 'or' c) '<' b 'or' c proof let z be Element of Y; A1: ((a 'or' b) '&' ('not' a 'or' c)).z =(a 'or' b).z '&' ('not' a 'or' c).z by MARGREL1:def 20 .=(a.z 'or' b.z) '&' ('not' a 'or' c).z by BVFUNC_1:def 4 .=(a.z 'or' b.z) '&' (('not' a).z 'or' c.z) by BVFUNC_1:def 4; assume A2: ((a 'or' b) '&' ('not' a 'or' c)).z=TRUE; now assume (b 'or' c).z<>TRUE; then (b 'or' c).z=FALSE by XBOOLEAN:def 3; then A3: b.z 'or' c.z=FALSE by BVFUNC_1:def 4; c.z=TRUE or c.z=FALSE by XBOOLEAN:def 3; then (a.z 'or' b.z) '&' (('not' a).z 'or' c.z) =a.z '&' 'not' a.z by A3, MARGREL1:def 19 .=FALSE by XBOOLEAN:138; hence contradiction by A2,A1; end; hence thesis; end; theorem Th918: (a 'imp' b) '&' (c 'imp' d) '<' (a '&' c) 'imp' (b '&' d) proof let z be Element of Y; A1: ((a 'imp' b) '&' (c 'imp' d)).z =(a 'imp' b).z '&' (c 'imp' d).z by MARGREL1:def 20 .=('not' a 'or' b).z '&' (c 'imp' d).z by BVFUNC_4:8 .=('not' a 'or' b).z '&' ('not' c 'or' d).z by BVFUNC_4:8 .=(('not' a).z 'or' b.z) '&' ('not' c 'or' d).z by BVFUNC_1:def 4 .=(('not' a).z 'or' b.z) '&' (('not' c).z 'or' (d).z) by BVFUNC_1:def 4; assume A2: ((a 'imp' b) '&' (c 'imp' d)).z=TRUE; now A3: ('not' c).z=TRUE or ('not' c).z=FALSE by XBOOLEAN:def 3; A4: (('not' a).z 'or' ('not' c).z)=TRUE or (('not' a).z 'or' ('not' c).z)= FALSE by XBOOLEAN:def 3; A5: (b.z '&' (d).z)=TRUE or (b.z '&' (d).z)=FALSE by XBOOLEAN:def 3; A6: ((a '&' c) 'imp' (b '&' d)).z =('not'( a '&' c) 'or' (b '&' d)).z by BVFUNC_4:8 .=('not'( a '&' c)).z 'or' (b '&' d).z by BVFUNC_1:def 4 .=('not' a 'or' 'not' c).z 'or' (b '&' d).z by BVFUNC_1:14 .=(('not' a).z 'or' ('not' c).z) 'or' (b '&' d).z by BVFUNC_1:def 4 .=(('not' a).z 'or' ('not' c).z) 'or' (b.z '&' (d).z) by MARGREL1:def 20; assume A7: ((a '&' c) 'imp' (b '&' d)).z<>TRUE; now per cases by A7,A6,A5,MARGREL1:12; case b.z=FALSE; thus thesis by A2,A1,A6,A4,A3; end; case d.z=FALSE; thus thesis by A2,A1,A6,A4,A3; end; end; hence thesis; end; hence thesis; end; theorem (a 'imp' b) '&' (a 'imp' c) '<' a 'imp' (b '&' c) proof (a 'imp' b) '&' (a 'imp' c) '<' (a '&' a) 'imp' (b '&' c) by Th918; hence thesis; end; theorem Th20: ((a 'imp' c) '&' (b 'imp' c)) '<' (a 'or' b) 'imp' c proof ((a 'imp' c) '&' (b 'imp' c)) 'imp' ((a 'or' b) 'imp' c) = I_el Y by Th9; hence thesis by BVFUNC_1:16; end; theorem Th21: (a 'imp' b) '&' (c 'imp' d) '<' (a 'or' c) 'imp' (b 'or' d) proof let z be Element of Y; A1: ((a 'imp' b) '&' (c 'imp' d)).z =(('not' a 'or' b) '&' (c 'imp' d)).z by BVFUNC_4:8 .=(('not' a 'or' b) '&' ('not' c 'or' d)).z by BVFUNC_4:8 .=('not' a 'or' b).z '&' ('not' c 'or' d).z by MARGREL1:def 20 .=(('not' a).z 'or' b.z) '&' ('not' c 'or' d).z by BVFUNC_1:def 4 .=(('not' a).z 'or' b.z) '&' (('not' c).z 'or' (d).z) by BVFUNC_1:def 4; assume A2: ((a 'imp' b) '&' (c 'imp' d)).z=TRUE; now A3: (d).z=TRUE or (d).z=FALSE by XBOOLEAN:def 3; A4: (b.z 'or' (d).z)=TRUE or (b.z 'or' (d).z)=FALSE by XBOOLEAN:def 3; A5: (('not' a).z '&' ('not' c).z)=TRUE or (('not' a).z '&' ('not' c).z)= FALSE by XBOOLEAN:def 3; A6: ((a 'or' c) 'imp' (b 'or' d)).z =('not'( a 'or' c) 'or' (b 'or' d)).z by BVFUNC_4:8 .=(('not' a '&' 'not' c) 'or' (b 'or' d)).z by BVFUNC_1:13 .=('not' a '&' 'not' c).z 'or' (b 'or' d).z by BVFUNC_1:def 4 .=(('not' a).z '&' ('not' c).z) 'or' (b 'or' d).z by MARGREL1:def 20 .=(('not' a).z '&' ('not' c).z) 'or' (b.z 'or' (d).z) by BVFUNC_1:def 4; assume A7: ((a 'or' c) 'imp' (b 'or' d)).z<>TRUE; now per cases by A7,A6,A5,MARGREL1:12; case ('not' a).z=FALSE; thus thesis by A2,A1,A6,A4,A3; end; case ('not' c).z=FALSE; thus thesis by A2,A1,A6,A4,A3; end; end; hence thesis; end; hence thesis; end; theorem (a 'imp' b) '&' (a 'imp' c) '<' a 'imp' (b 'or' c) proof (a 'imp' b) '&' (a 'imp' c) '<' (a 'or' a) 'imp' (b 'or' c) by Th21; hence thesis; end; theorem Th23: for a1,b1,c1,a2,b2,c2 being Function of Y,BOOLEAN holds ( b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '<' (a2 'imp' a1) proof let a1,b1,c1,a2,b2,c2 be Function of Y,BOOLEAN; A1: ((b1 'or' c1) 'imp' (b2 'or' c2)) '&' ((b2 'or' c2) 'imp' 'not' a2) '&' ((b1 'or' c1) 'imp' 'not' a2) 'imp' ((b1 'or' c1) 'imp' 'not' a2) = I_el(Y) by Th38; A2: (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) 'imp' (c1 'imp' c2) = I_el(Y) by Lm4,BVFUNC_1:16; A3: ((b1 'imp' b2) '&' (c1 'imp' c2)) 'imp' ((b1 'or' c1) 'imp' (b2 'or' c2 )) = I_el(Y) by Th21,BVFUNC_1:16; (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) 'imp' (b1 'imp' b2) = I_el(Y) by Lm4,BVFUNC_1:16; then (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) 'imp' (b1 'imp' b2) '&' (c1 'imp' c2) = I_el(Y) by A2,th18; then A4: (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) 'imp' ((b1 'or' c1) 'imp' (b2 'or' c2)) = I_el(Y) by A3,BVFUNC_5:9; A5: (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) 'imp' 'not'( a2 '&' c2) = I_el(Y) by Lm1, BVFUNC_1:16; (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) 'imp' 'not'( a2 '&' b2) = I_el(Y) by Lm2, BVFUNC_1:16; then (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) 'imp' ('not'( a2 '&' b2) '&' 'not'( a2 '&' c2)) = I_el(Y) by A5,th18; then A6: (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) 'imp' ((b1 'or' c1) 'imp' (b2 'or' c2)) '&' ( 'not'( a2 '&' b2) '&' 'not' (a2 '&' c2)) = I_el(Y) by A4,th18; 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) =('not' a2 'or' 'not' b2) '&' 'not'( a2 '&' c2) by BVFUNC_1:14 .=('not' b2 'or' 'not' a2) '&' ('not' c2 'or' 'not' a2) by BVFUNC_1:14 .=(b2 'imp' 'not' a2) '&' ('not' c2 'or' 'not' a2) by BVFUNC_4:8 .=(b2 'imp' 'not' a2) '&' (c2 'imp' 'not' a2) by BVFUNC_4:8 .=(b2 'or' c2) 'imp' 'not' a2 by Th75; then (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) 'imp' ((b1 'or' c1) 'imp' (b2 'or' c2)) '&' ((b2 'or' c2) 'imp' 'not' a2) '&' ((b1 'or' c1) 'imp' 'not' a2) = I_el(Y) by A6, Th12; then A7: (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) 'imp' ((b1 'or' c1) 'imp' 'not' a2) = I_el(Y) by A1,BVFUNC_5:9; (a1 'or' b1 'or' c1) '&' ((b1 'or' c1) 'imp' 'not' a2) =(a1 'or' (b1 'or' c1) ) '&' ((b1 'or' c1) 'imp' 'not' a2) & (a1 'or' (b1 'or' c1)) '&' ((b1 'or' c1) 'imp' 'not' a2) '<' (a1 'or' 'not' a2) by Th1,BVFUNC_1:8; then A8: (a1 'or' b1 'or' c1) '&' ((b1 'or' c1) 'imp' 'not' a2) 'imp' (a1 'or' 'not' a2) =I_el(Y) by BVFUNC_1:16; (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) 'imp' (a1 'or' b1 'or' c1) = I_el(Y) by Lm3, BVFUNC_1:16; then (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) 'imp' (a1 'or' b1 'or' c1) '&' ((b1 'or' c1) 'imp' 'not' a2) = I_el(Y) by A7,th18; then (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) 'imp' (a1 'or' 'not' a2) = I_el(Y) by A8, BVFUNC_5:9; then (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) 'imp' (a2 'imp' a1) = I_el(Y) by BVFUNC_4:8; hence thesis by BVFUNC_1:16; end; Lm7: for a1,b1,c1,a2,b2,c2 being Function of Y,BOOLEAN holds ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ((a1 'imp' a2) '&' ( b1 'imp' b2) '&' (a1 'or' b1 'or' c1) '&' 'not'( c2 '&' a2) '&' 'not'( c2 '&' b2)) =I_el(Y) proof let a1,b1,c1,a2,b2,c2 be Function of Y,BOOLEAN; A1: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' 'not'( a2 '&' c2) = I_el(Y) by Lm2,BVFUNC_1:16; ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ( a1 'imp' a2) = I_el(Y) & ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' ( a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (b1 'imp' b2) = I_el(Y) by Lm6,BVFUNC_1:16; then A2: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ( a1 'imp' a2) '&' (b1 'imp' b2) = I_el(Y) by th18; ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ( a1 'or' b1 'or' c1) = I_el(Y) by Lm4,BVFUNC_1:16; then ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ( a1 'imp' a2) '&' (b1 'imp' b2) '&' (a1 'or' b1 'or' c1) = I_el(Y) by A2, th18; then A3: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ( a1 'imp' a2) '&' (b1 'imp' b2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' c2) = I_el(Y) by A1,th18; ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' 'not'( b2 '&' c2) = I_el(Y) by Lm1,BVFUNC_1:16; hence thesis by A3,th18; end; Lm8: for a1,b1,c1,a2,b2,c2 being Function of Y,BOOLEAN holds ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ((a1 'imp' a2) '&' ( c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( b2 '&' a2) '&' 'not'( b2 '&' c2)) = I_el(Y) proof let a1,b1,c1,a2,b2,c2 be Function of Y,BOOLEAN; A1: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' 'not'( a2 '&' b2) = I_el(Y) by Lm3,BVFUNC_1:16; ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ( a1 'imp' a2) = I_el(Y) & ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' ( a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (c1 'imp' c2) = I_el(Y) by Lm5,Lm6,BVFUNC_1:16; then A2: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ( a1 'imp' a2) '&' (c1 'imp' c2) = I_el(Y) by th18; ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ( a1 'or' b1 'or' c1) = I_el(Y) by Lm4,BVFUNC_1:16; then ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ( a1 'imp' a2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) = I_el(Y) by A2, th18; then A3: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ( a1 'imp' a2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) = I_el(Y) by A1,th18; ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' 'not'( b2 '&' c2) = I_el(Y) by Lm1,BVFUNC_1:16; hence thesis by A3,th18; end; Lm9: for a1,b1,c1,a2,b2,c2 being Function of Y,BOOLEAN holds ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ((b1 'imp' b2) '&' ( c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2)) =I_el(Y) proof let a1,b1,c1,a2,b2,c2 be Function of Y,BOOLEAN; A1: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' 'not'( a2 '&' b2) = I_el(Y) by Lm3,BVFUNC_1:16; ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ( b1 'imp' b2) = I_el(Y) & ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' ( a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (c1 'imp' c2) = I_el(Y) by Lm5,Lm6,BVFUNC_1:16; then A2: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ( b1 'imp' b2) '&' (c1 'imp' c2) = I_el(Y) by th18; ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ( a1 'or' b1 'or' c1) = I_el(Y) by Lm4,BVFUNC_1:16; then ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ( b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) = I_el(Y) by A2, th18; then A3: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ( b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) = I_el(Y) by A1,th18; ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' 'not'( a2 '&' c2) = I_el(Y) by Lm2,BVFUNC_1:16; hence thesis by A3,th18; end; theorem for a1,b1,c1,a2,b2,c2 being Function of Y,BOOLEAN holds (a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2) '<' (a2 'imp' a1) '&' (b2 'imp' b1) '&' (c2 'imp' c1) proof let a1,b1,c1,a2,b2,c2 be Function of Y,BOOLEAN; A1: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (( b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2)) =I_el(Y) by Lm9; ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (( a1 'imp' a2) '&' (b1 'imp' b2) '&' (a1 'or' b1 'or' c1) '&' 'not'( c2 '&' a2) '&' 'not'( c2 '&' b2)) =I_el(Y) & ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( b2 '&' a2) '&' 'not'( b2 '&' c2)) =I_el(Y) by Lm7,Lm8; then ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (( a1 'imp' a2) '&' (b1 'imp' b2) '&' (a1 'or' b1 'or' c1) '&' 'not'( c2 '&' a2) '&' 'not'( c2 '&' b2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( b2 '&' a2) '&' 'not'( b2 '&' c2)) =I_el(Y) by th18; then A2: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (( a1 'imp' a2) '&' (b1 'imp' b2) '&' (a1 'or' b1 'or' c1) '&' 'not'( c2 '&' a2) '&' 'not'( c2 '&' b2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( b2 '&' c2)) '&' ((b1 'imp' b2) '&' ( c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2)) =I_el(Y) by A1,th18; A3: (a1 'imp' a2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( b2 '&' c2) 'imp' (b2 'imp' b1) = I_el(Y) by Th23,BVFUNC_1:16; ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (a1 'or' b1 'or' c1) '&' 'not'( c2 '&' a2) '&' 'not'( c2 '&' b2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( b2 '&' c2)) '&' ((b1 'imp' b2 ) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2)) 'imp' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( b2 '&' c2)) =I_el(Y) by Lm2,BVFUNC_1:16; then ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (( a1 'imp' a2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( b2 '&' c2)) =I_el(Y) by A2,BVFUNC_5:9; then A4: (a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2) 'imp' (b2 'imp' b1) = I_el(Y) by A3,BVFUNC_5:9; (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) 'imp' (a2 'imp' a1) = I_el(Y) by Th23,BVFUNC_1:16; then (a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not' (b2 '&' c2) 'imp' (a2 'imp' a1) = I_el(Y) by A1,BVFUNC_5:9; then A5: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ( a2 'imp' a1) '&' (b2 'imp' b1) = I_el(Y) by A4,th18; (a1 'imp' a2) '&' (b1 'imp' b2) '&' (a1 'or' c1 'or' b1) '&' 'not'( c2 '&' a2) '&' 'not'( c2 '&' b2) '<' (c2 'imp' c1) by Th23; then (a1 'imp' a2) '&' (b1 'imp' b2) '&' (a1 'or' b1 'or' c1) '&' 'not'( c2 '&' a2) '&' 'not'( c2 '&' b2) '<' (c2 'imp' c1) by BVFUNC_1:8; then A6: (a1 'imp' a2) '&' (b1 'imp' b2) '&' (a1 'or' b1 'or' c1) '&' 'not'( c2 '&' a2) '&' 'not'( c2 '&' b2) 'imp' (c2 'imp' c1) = I_el(Y) by BVFUNC_1:16; ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (a1 'or' b1 'or' c1) '&' 'not'( c2 '&' a2) '&' 'not'( c2 '&' b2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( b2 '&' c2)) '&' ((b1 'imp' b2 ) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2)) 'imp' ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (a1 'or' b1 'or' c1) '&' 'not'( c2 '&' a2) '&' 'not'( c2 '&' b2)) =I_el(Y) by Lm2,BVFUNC_1:16; then ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' (( a1 'imp' a2) '&' (b1 'imp' b2) '&' (a1 'or' b1 'or' c1) '&' 'not'( c2 '&' a2) '&' 'not'( c2 '&' b2)) =I_el(Y) by A2,BVFUNC_5:9; then (a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2) 'imp' (c2 'imp' c1) = I_el(Y) by A6,BVFUNC_5:9; then ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2)) 'imp' ( a2 'imp' a1) '&' (b2 'imp' b1) '&' (c2 'imp' c1) = I_el(Y) by A5,th18; hence thesis by BVFUNC_1:16; end; theorem Th25: for a1,b1,a2,b2 being Function of Y,BOOLEAN holds ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'(a2 '&' b2)) 'imp' 'not' (a1 '&' b1)=I_el( Y) proof let a1,b1,a2,b2 be Function of Y,BOOLEAN; for z being Element of Y st ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)).z=TRUE holds ('not'( a1 '&' b1)).z=TRUE proof let z be Element of Y; A1: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)).z =((a1 'imp' a2) '&' (b1 'imp' b2)).z '&' ('not'( a2 '&' b2)).z by MARGREL1:def 20 .=((a1 'imp' a2).z '&' (b1 'imp' b2).z) '&' ('not'( a2 '&' b2)).z by MARGREL1:def 20 .=(('not' a1 'or' a2).z '&' (b1 'imp' b2).z) '&' ('not'( a2 '&' b2)).z by BVFUNC_4:8 .=(('not' a1 'or' a2).z '&' ('not' b1 'or' b2).z) '&' ('not' (a2 '&' b2)).z by BVFUNC_4:8 .=((('not' a1).z 'or' (a2).z) '&' ('not' b1 'or' b2).z) '&' ('not'( a2 '&' b2)).z by BVFUNC_1:def 4 .=((('not' a1).z 'or' (a2).z) '&' (('not' b1).z 'or' (b2).z)) '&' ( 'not'( a2 '&' b2)).z by BVFUNC_1:def 4 .=((('not' a1).z 'or' (a2).z) '&' (('not' b1).z 'or' (b2).z)) '&' (( 'not' a2 'or' 'not' b2).z) by BVFUNC_1:14 .=((('not' a1).z 'or' (a2).z) '&' (('not' b1).z 'or' (b2).z)) '&' (( 'not' a2).z 'or' ('not' b2).z) by BVFUNC_1:def 4; assume A2: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)).z=TRUE; now A3: ('not' b1).z=TRUE or ('not' b1).z=FALSE by XBOOLEAN:def 3; A4: ('not' a1).z=TRUE or ('not' a1).z=FALSE by XBOOLEAN:def 3; A5: ('not'( a1 '&' b1)).z =('not' a1 'or' 'not' b1).z by BVFUNC_1:14 .=('not' a1).z 'or' ('not' b1).z by BVFUNC_1:def 4; assume ('not'( a1 '&' b1)).z<>TRUE; then ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)).z =((b2).z '&' (a2).z) '&' ('not' a2).z 'or' ((a2).z '&' (b2).z) '&' ('not' b2).z by A1,A5 ,A4,A3,XBOOLEAN:8 .=b2.z '&' ((a2).z '&' ('not' a2).z) 'or' (a2).z '&' ((b2).z '&' ( 'not' b2).z) .=b2.z '&' ((a2).z '&' 'not' (a2).z) 'or' (a2).z '&' ((b2).z '&' ( 'not' b2).z) by MARGREL1:def 19 .=b2.z '&' ((a2).z '&' 'not' (a2).z) 'or' (a2).z '&' ((b2).z '&' 'not' (b2).z) by MARGREL1:def 19 .=b2.z '&' FALSE 'or' (a2).z '&' ((b2).z '&' 'not' (b2).z) by XBOOLEAN:138 .=FALSE 'or' FALSE '&' a2.z by XBOOLEAN:138 .=FALSE; hence contradiction by A2; end; hence thesis; end; hence thesis by BVFUNC_1:16,def 12; end; theorem for a1,b1,c1,a2,b2,c2 being Function of Y,BOOLEAN holds (a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2) '<' 'not'( a1 '&' b1) '&' 'not'( a1 '&' c1) '&' 'not'( b1 '&' c1) proof let a1,b1,c1,a2,b2,c2 be Function of Y,BOOLEAN; A1: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2) '&' 'not'( b2 '&' c2)) 'imp' ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) =I_el(Y) by Th38; A2: ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) 'imp' 'not'( a1 '&' c1)=I_el(Y) by Th25; ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) 'imp' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) =I_el(Y) by Th38; then ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2) '&' 'not'( b2 '&' c2)) 'imp' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) =I_el(Y) by A1,BVFUNC_5:9; then A3: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2) '&' 'not'( b2 '&' c2)) 'imp' 'not'( a1 '&' c1)=I_el(Y) by A2,BVFUNC_5:9; A4: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) 'imp' 'not'( a1 '&' b1)=I_el(Y) by Th25; ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2) '&' 'not'( b2 '&' c2)) 'imp' ((b1 'imp' b2) '&' (c1 'imp' c2) '&' 'not'( b2 '&' c2)) =I_el(Y) & ((b1 'imp' b2) '&' (c1 'imp' c2) '&' 'not'( b2 '&' c2)) 'imp' 'not'( b1 '&' c1)=I_el(Y) by Th25,Th38; then A5: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2) '&' 'not'( b2 '&' c2)) 'imp' 'not'( b1 '&' c1)=I_el(Y) by BVFUNC_5:9; ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) 'imp' ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) =I_el(Y) by Th38; then ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2) '&' 'not'( b2 '&' c2)) 'imp' ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) =I_el(Y) by A1,BVFUNC_5:9; then ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2) '&' 'not'( b2 '&' c2)) 'imp' 'not'(a1 '&' b1)=I_el(Y) by A4,BVFUNC_5:9; then ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not'(a2 '&' c2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2) '&' 'not'( b2 '&' c2)) 'imp' 'not'( a1 '&' b1) '&' 'not'( a1 '&' c1) =I_el( Y) by A3,th18; then A6: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2) '&' 'not'( b2 '&' c2)) 'imp' 'not'( a1 '&' b1) '&' 'not'( a1 '&' c1) '&' 'not'( b1 '&' c1) =I_el(Y) by A5,th18; (a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2) =(a1 'imp' a2) '&' (a1 'imp' a2) '&' ((b1 'imp' b2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2) .=(((a1 'imp' a2) '&' (a1 'imp' a2) '&' (b1 'imp' b2)) '&' (b1 'imp' b2) '&' ((c1 'imp' c2) '&' (c1 'imp' c2))) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2) by BVFUNC_1:4 .=((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (a1 'imp' a2)) '&' (b1 'imp' b2) '&' ((c1 'imp' c2) '&' (c1 'imp' c2))) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2) by BVFUNC_1:4 .=(((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (c1 'imp' c2)) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2 ) '&' 'not'( b2 '&' c2) by BVFUNC_1:4 .=(((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (a1 'imp' a2) '&' ((b1 'imp' b2 ) '&' (c1 'imp' c2)) '&' (c1 'imp' c2)) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2) by BVFUNC_1:4 .=(((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2 )) '&' (a1 'imp' a2) '&' (c1 'imp' c2)) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2) by BVFUNC_1:4 .=((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2) ) '&' ((a1 'imp' a2) '&' (c1 'imp' c2)) '&' 'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) '&' 'not'( b2 '&' c2) by BVFUNC_1:4 .=((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2) ) '&' ((a1 'imp' a2) '&' (c1 'imp' c2)) '&' 'not'( a2 '&' c2) '&' 'not'( a2 '&' b2) '&' 'not'( b2 '&' c2) by BVFUNC_1:4 .=((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2) ) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) '&' 'not'( a2 '&' b2) '&' 'not'( b2 '&' c2) by BVFUNC_1:4 .=((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2) ) '&' 'not'( a2 '&' b2) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not' (a2 '&' c2)) '&' 'not'( b2 '&' c2) by BVFUNC_1:4 .=((a1 'imp' a2) '&' (b1 'imp' b2)) '&' 'not'( a2 '&' b2) '&' ((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) '&' 'not'( b2 '&' c2) by BVFUNC_1:4 .=((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2)) '&' 'not'( b2 '&' c2) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not' (a2 '&' c2)) by BVFUNC_1:4 .=((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2) '&' 'not'( b2 '&' c2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) by BVFUNC_1:4 .=(((a1 'imp' a2) '&' (b1 'imp' b2) '&' 'not'( a2 '&' b2)) '&' ((a1 'imp' a2) '&' (c1 'imp' c2) '&' 'not'( a2 '&' c2)) '&' ((b1 'imp' b2) '&' (c1 'imp' c2) '&' 'not'( b2 '&' c2))) by BVFUNC_1:4; hence thesis by A6,BVFUNC_1:16; end; theorem a '&' b '<' a by Lm1; theorem a '&' b '&' c '<' a & a '&' b '&' c '<' b by Lm2; theorem a '&' b '&' c '&' d '<' a & a '&' b '&' c '&' d '<' b by Lm3; theorem a '&' b '&' c '&' d '&' e '<' a & a '&' b '&' c '&' d '&' e '<' b by Lm4; theorem a '&' b '&' c '&' d '&' e '&' f '<' a & a '&' b '&' c '&' d '&' e '&' f '<' b by Lm5; theorem a '&' b '&' c '&' d '&' e '&' f '&' g '<' a & a '&' b '&' c '&' d '&' e '&' f '&' g '<' b by Lm6; theorem Th33: a '<' b & c '<' d implies a '&' c '<' b '&' d proof assume a '<' b & c '<' d; then a 'imp' b = I_el Y & c 'imp' d = I_el Y by BVFUNC_1:16; then (a '&' c) 'imp' (b '&' d) = I_el Y by tt; hence thesis by BVFUNC_1:16; end; theorem a '&' b '<' c implies a '&' 'not' c '<' 'not' b proof assume a '&' b '<' c; then I_el Y = a '&' b 'imp' c by BVFUNC_1:16 .= 'not' (a '&' b) 'or' c by BVFUNC_4:8 .= 'not' a 'or' 'not' b 'or' c by BVFUNC_1:14 .= 'not' a 'or' 'not' 'not' c 'or' 'not' b by BVFUNC_1:8 .= 'not' (a '&' 'not' c) 'or' 'not' b by BVFUNC_1:14 .= a '&' 'not' c 'imp' 'not' b by BVFUNC_4:8; hence thesis by BVFUNC_1:16; end; theorem (a 'imp' c) '&' (b 'imp' c) '&' (a 'or' b) '<' c proof set K1 = (a 'imp' c) '&' (b 'imp' c); K1 '<' (a 'or' b) 'imp' c by Th20; then A1: K1 '&' (a 'or' b) '<' ((a 'or' b) 'imp' c) '&' (a 'or' b) by Th33; ((a 'or' b) 'imp' c) '&' (a 'or' b) '<' c by Th2; hence thesis by A1,BVFUNC_1:15; end; theorem ((a 'imp' c) 'or' (b 'imp' c)) '&' (a '&' b) '<' c proof (a 'imp' c) 'or' (b 'imp' c) = (a '&' b) 'imp' c by Th76; hence thesis by Th2; end; theorem a '<' b & c '<' d implies a 'or' c '<' b 'or' d proof assume a '<' b & c '<' d; then a 'imp' b = I_el(Y) & c 'imp' d = I_el(Y) by BVFUNC_1:16; then (a 'or' c) 'imp' (b 'or' d) = I_el(Y) by Th22; hence thesis by BVFUNC_1:16; end; theorem Th38a: a '<' a 'or' b proof a 'imp' (a 'or' b) = I_el Y by Th26; hence thesis by BVFUNC_1:16; end; theorem a '&' b '<' a 'or' b proof a '&' b '<' a & a '<' a 'or' b by Lm1,Th38a; hence thesis by BVFUNC_1:15; end; begin :: BVFUNC10 reserve Y for non empty set; theorem for a,b,c being Function of Y,BOOLEAN holds (a '&' b) 'or' (b '&' c) 'or' (c '&' a)= (a 'or' b) '&' (b 'or' c) '&' (c 'or' a) proof let a,b,c be Function of Y,BOOLEAN; for z being Element of Y st ((a '&' b) 'or' (b '&' c) 'or' (c '&' a)).z= TRUE holds ((a 'or' b) '&' (b 'or' c) '&' (c 'or' a)).z=TRUE proof let z be Element of Y; A1: ((a '&' b) 'or' (b '&' c) 'or' (c '&' a)).z =((a '&' b) 'or' (b '&' c) ).z 'or' (c '&' a).z by BVFUNC_1:def 4 .=(a '&' b).z 'or' (b '&' c).z 'or' (c '&' a).z by BVFUNC_1:def 4 .=(a.z '&' b.z) 'or' (b '&' c).z 'or' (c '&' a).z by MARGREL1:def 20 .=(a.z '&' b.z) 'or' (b.z '&' c.z) 'or' (c '&' a).z by MARGREL1:def 20 .=(a.z '&' b.z) 'or' (b.z '&' c.z) 'or' (c.z '&' a.z) by MARGREL1:def 20; assume A2: ((a '&' b) 'or' (b '&' c) 'or' (c '&' a)).z=TRUE; now A3: ((a 'or' b) '&' (b 'or' c) '&' (c 'or' a)).z =((a 'or' b) '&' (b 'or' c)).z '&' (c 'or' a).z by MARGREL1:def 20 .=(a 'or' b).z '&' (b 'or' c).z '&' (c 'or' a).z by MARGREL1:def 20 .=(a.z 'or' b.z) '&' (b 'or' c).z '&' (c 'or' a).z by BVFUNC_1:def 4 .=(a.z 'or' b.z) '&' (b.z 'or' c.z) '&' (c 'or' a).z by BVFUNC_1:def 4 .=(a.z 'or' b.z) '&' (b.z 'or' c.z) '&' (c.z 'or' a.z) by BVFUNC_1:def 4; assume A4: ((a 'or' b) '&' (b 'or' c) '&' (c 'or' a)).z<>TRUE; now per cases by A4,A3,MARGREL1:12,XBOOLEAN:def 3; case A5: (a.z 'or' b.z) '&' (b.z 'or' c.z)=FALSE; now per cases by A5,MARGREL1:12; case A6: (a.z 'or' b.z)=FALSE; b.z=TRUE or b.z=FALSE by XBOOLEAN:def 3; hence thesis by A2,A1,A6; end; case A7: (b.z 'or' c.z)=FALSE; c.z=TRUE or c.z=FALSE by XBOOLEAN:def 3; hence thesis by A2,A1,A7; end; end; hence thesis; end; case A8: (c.z 'or' a.z)=FALSE; a.z=TRUE or a.z=FALSE by XBOOLEAN:def 3; hence thesis by A2,A1,A8; end; end; hence thesis; end; hence thesis; end; then A9: (a '&' b) 'or' (b '&' c) 'or' (c '&' a) '<' (a 'or' b) '&' (b 'or' c) '&' (c 'or' a); for z being Element of Y st ((a 'or' b) '&' (b 'or' c) '&' (c 'or' a)). z=TRUE holds ((a '&' b) 'or' (b '&' c) 'or' (c '&' a)).z=TRUE proof let z be Element of Y; A10: ((a 'or' b) '&' (b 'or' c) '&' (c 'or' a)).z =((a 'or' b) '&' (b 'or' c)).z '&' (c 'or' a).z by MARGREL1:def 20 .=(a 'or' b).z '&' (b 'or' c).z '&' (c 'or' a).z by MARGREL1:def 20 .=(a.z 'or' b.z) '&' (b 'or' c).z '&' (c 'or' a).z by BVFUNC_1:def 4 .=(a.z 'or' b.z) '&' (b.z 'or' c.z) '&' (c 'or' a).z by BVFUNC_1:def 4 .=(a.z 'or' b.z) '&' (b.z 'or' c.z) '&' (c.z 'or' a.z) by BVFUNC_1:def 4; assume A11: ((a 'or' b) '&' (b 'or' c) '&' (c 'or' a)).z=TRUE; now A12: (b.z '&' c.z)=TRUE or (b.z '&' c.z)=FALSE by XBOOLEAN:def 3; A13: (c.z '&' a.z)=TRUE or (c.z '&' a.z)=FALSE by XBOOLEAN:def 3; A14: ((a '&' b) 'or' (b '&' c) 'or' (c '&' a)).z =((a '&' b) 'or' (b '&' c)).z 'or' (c '&' a).z by BVFUNC_1:def 4 .=(a '&' b).z 'or' (b '&' c).z 'or' (c '&' a).z by BVFUNC_1:def 4 .=(a.z '&' b.z) 'or' (b '&' c).z 'or' (c '&' a).z by MARGREL1:def 20 .=(a.z '&' b.z) 'or' (b.z '&' c.z) 'or' (c '&' a).z by MARGREL1:def 20 .=(a.z '&' b.z) 'or' (b.z '&' c.z) 'or' (c.z '&' a.z) by MARGREL1:def 20; assume A15: ((a '&' b) 'or' (b '&' c) 'or' (c '&' a)).z<>TRUE; now per cases by A15,A14,A13,A12,MARGREL1:12,XBOOLEAN:def 3; case a.z=FALSE & b.z=FALSE; hence thesis by A11,A10; end; case b.z=FALSE & c.z=FALSE; hence thesis by A11,A10; end; case c.z=FALSE & a.z=FALSE; hence thesis by A11,A10; end; end; hence thesis; end; hence thesis; end; then (a 'or' b) '&' (b 'or' c) '&' (c 'or' a) '<' (a '&' b) 'or' (b '&' c) 'or' (c '&' a); hence thesis by A9,BVFUNC_1:15; end; Lm1: for a,b,c being Function of Y,BOOLEAN holds (a '&' 'not' b) 'or' (b '&' 'not' c) 'or' (c '&' 'not' a) '<' (b '&' 'not' a) 'or' (c '&' 'not' b) 'or' (a '&' 'not' c) proof let a,b,c be Function of Y,BOOLEAN; let z be Element of Y; A1: ((a '&' 'not' b) 'or' (b '&' 'not' c) 'or' (c '&' 'not' a)).z =((a '&' 'not' b) 'or' (b '&' 'not' c)).z 'or' (c '&' 'not' a).z by BVFUNC_1:def 4 .=(a '&' 'not' b).z 'or' (b '&' 'not' c).z 'or' (c '&' 'not' a).z by BVFUNC_1:def 4 .=(a.z '&' ('not' b).z) 'or' (b '&' 'not' c).z 'or' (c '&' 'not' a).z by MARGREL1:def 20 .=(a.z '&' ('not' b).z) 'or' (b.z '&' ('not' c).z) 'or' (c '&' 'not' a).z by MARGREL1:def 20 .=(a.z '&' ('not' b).z) 'or' (b.z '&' ('not' c).z) 'or' (c.z '&' ( 'not' a).z) by MARGREL1:def 20 .=(a.z '&' 'not' b.z) 'or' (b.z '&' ('not' c).z) 'or' (c.z '&' ( 'not' a).z) by MARGREL1:def 19 .=(a.z '&' 'not' b.z) 'or' (b.z '&' 'not' c.z) 'or' (c.z '&' ( 'not' a).z) by MARGREL1:def 19 .=(a.z '&' 'not' b.z) 'or' (b.z '&' 'not' c.z) 'or' (c.z '&' 'not' a.z) by MARGREL1:def 19; assume A2: ((a '&' 'not' b) 'or' (b '&' 'not' c) 'or' (c '&' 'not' a)).z=TRUE; now A3: (a.z '&' ('not' c).z)=TRUE or (a.z '&' ('not' c).z)=FALSE by XBOOLEAN:def 3; assume A4: ((b '&' 'not' a) 'or' (c '&' 'not' b) 'or' (a '&' 'not' c)).z<> TRUE; A5: (c.z '&' ('not' b).z)=TRUE or (c.z '&' ('not' b).z)=FALSE by XBOOLEAN:def 3; A6: ((b '&' 'not' a) 'or' (c '&' 'not' b) 'or' (a '&' 'not' c)).z =((b '&' 'not' a) 'or' (c '&' 'not' b)).z 'or' (a '&' 'not' c).z by BVFUNC_1:def 4 .=(b '&' 'not' a).z 'or' (c '&' 'not' b).z 'or' (a '&' 'not' c).z by BVFUNC_1:def 4 .=(b.z '&' ('not' a).z) 'or' (c '&' 'not' b).z 'or' (a '&' 'not' c). z by MARGREL1:def 20 .=(b.z '&' ('not' a).z) 'or' (c.z '&' ('not' b).z) 'or' (a '&' 'not' c).z by MARGREL1:def 20 .=(b.z '&' ('not' a).z) 'or' (c.z '&' ('not' b).z) 'or' (a.z '&' ('not' c).z) by MARGREL1:def 20; ((b.z '&' ('not' a).z) 'or' (c.z '&' ('not' b).z))=TRUE or ((b.z '&' ('not' a).z) 'or' (c.z '&' ('not' b).z))=FALSE by XBOOLEAN:def 3; then A7: b.z=FALSE or ('not' a).z=FALSE by A4,A6,A5,MARGREL1:12; now per cases by A4,A6,A3,MARGREL1:12; case a.z=FALSE; hence thesis by A2,A1,A6,A7,MARGREL1:def 19; end; case ('not' c).z=FALSE; then A8: 'not' c.z=FALSE by MARGREL1:def 19; then 'not' b.z=FALSE by A4,A6,A5,MARGREL1:def 19; hence thesis by A2,A1,A6,A5,A8,MARGREL1:def 19; end; end; hence thesis; end; hence thesis; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a '&' 'not' b) 'or' (b '&' 'not' c) 'or' (c '&' 'not' a)= (b '&' 'not' a) 'or' (c '&' 'not' b) 'or' (a '&' 'not' c) proof let a,b,c be Function of Y,BOOLEAN; (a '&' 'not' b) 'or' (b '&' 'not' c) 'or' (c '&' 'not' a) '<' (b '&' 'not' a) 'or' (c '&' 'not' b) 'or' (a '&' 'not' c) & (b '&' 'not' a) 'or' (c '&' 'not' b) 'or' (a '&' 'not' c) '<' (a '&' 'not' b) 'or' (b '&' 'not' c) 'or' (c '&' 'not' a) by Lm1; hence thesis by BVFUNC_1:15; end; Lm2: for a,b,c being Function of Y,BOOLEAN holds (a 'or' 'not' b) '&' (b 'or' 'not' c) '&' (c 'or' 'not' a) '<' (b 'or' 'not' a) '&' (c 'or' 'not' b) '&' (a 'or' 'not' c) proof let a,b,c be Function of Y,BOOLEAN; let z be Element of Y; A1: ((a 'or' 'not' b) '&' (b 'or' 'not' c) '&' (c 'or' 'not' a)).z =((a 'or' 'not' b) '&' (b 'or' 'not' c)).z '&' (c 'or' 'not' a).z by MARGREL1:def 20 .=(a 'or' 'not' b).z '&' (b 'or' 'not' c).z '&' (c 'or' 'not' a).z by MARGREL1:def 20 .=(a.z 'or' ('not' b).z) '&' (b 'or' 'not' c).z '&' (c 'or' 'not' a).z by BVFUNC_1:def 4 .=(a.z 'or' ('not' b).z) '&' (b.z 'or' ('not' c).z) '&' (c 'or' 'not' a).z by BVFUNC_1:def 4 .=(a.z 'or' ('not' b).z) '&' (b.z 'or' ('not' c).z) '&' (c.z 'or' ('not' a).z) by BVFUNC_1:def 4 .=(a.z 'or' 'not' b.z) '&' (b.z 'or' ('not' c).z) '&' (c.z 'or' ('not' a).z) by MARGREL1:def 19 .=(a.z 'or' 'not' b.z) '&' (b.z 'or' 'not' c.z) '&' (c.z 'or' ('not' a).z) by MARGREL1:def 19 .=(a.z 'or' 'not' b.z) '&' (b.z 'or' 'not' c.z) '&' (c.z 'or' 'not' a.z) by MARGREL1:def 19; assume A2: ((a 'or' 'not' b) '&' (b 'or' 'not' c) '&' (c 'or' 'not' a)).z=TRUE; now A3: ((b 'or' 'not' a) '&' (c 'or' 'not' b) '&' (a 'or' 'not' c)).z =((b 'or' 'not' a) '&' (c 'or' 'not' b)).z '&' (a 'or' 'not' c).z by MARGREL1:def 20 .=(b 'or' 'not' a).z '&' (c 'or' 'not' b).z '&' (a 'or' 'not' c).z by MARGREL1:def 20 .=(b.z 'or' ('not' a).z) '&' (c 'or' 'not' b).z '&' (a 'or' 'not' c) .z by BVFUNC_1:def 4 .=(b.z 'or' ('not' a).z) '&' (c.z 'or' ('not' b).z) '&' (a 'or' 'not' c).z by BVFUNC_1:def 4 .=(b.z 'or' ('not' a).z) '&' (c.z 'or' ('not' b).z) '&' (a.z 'or' ('not' c).z) by BVFUNC_1:def 4; assume A4: ((b 'or' 'not' a) '&' (c 'or' 'not' b) '&' (a 'or' 'not' c)).z<> TRUE; now per cases by A4,A3,MARGREL1:12,XBOOLEAN:def 3; case A5: ((b.z 'or' ('not' a).z) '&' (c.z 'or' ('not' b).z))=FALSE; now per cases by A5,MARGREL1:12; case A6: (b.z 'or' ('not' a).z)=FALSE; A7: ('not' a).z=TRUE or ('not' a).z=FALSE by XBOOLEAN:def 3; then 'not' a.z=FALSE by A6,MARGREL1:def 19; hence thesis by A2,A1,A6,A7,XBOOLEAN:138; end; case A8: (c.z 'or' ('not' b).z)=FALSE; A9: ('not' b).z=TRUE or ('not' b).z=FALSE by XBOOLEAN:def 3; then 'not' b.z=FALSE by A8,MARGREL1:def 19; hence thesis by A2,A1,A8,A9,XBOOLEAN:138; end; end; hence thesis; end; case A10: (a.z 'or' ('not' c).z)=FALSE; A11: ('not' c).z=TRUE or ('not' c).z=FALSE by XBOOLEAN:def 3; then 'not' c.z=FALSE by A10,MARGREL1:def 19; hence thesis by A2,A1,A10,A11,XBOOLEAN:138; end; end; hence thesis; end; hence thesis; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'or' 'not' b) '&' (b 'or' 'not' c) '&' (c 'or' 'not' a)= (b 'or' 'not' a) '&' (c 'or' 'not' b) '&' (a 'or' 'not' c) proof let a,b,c be Function of Y,BOOLEAN; (a 'or' 'not' b) '&' (b 'or' 'not' c) '&' (c 'or' 'not' a) '<' (b 'or' 'not' a) '&' (c 'or' 'not' b) '&' (a 'or' 'not' c) & (b 'or' 'not' a) '&' (c 'or' 'not' b) '&' (a 'or' 'not' c) '<' (a 'or' 'not' b) '&' (b 'or' 'not' c) '&' (c 'or' 'not' a) by Lm2; hence thesis by BVFUNC_1:15; end; theorem for a,b,c being Function of Y,BOOLEAN holds (c 'imp' a)=I_el(Y) & (c 'imp' b)=I_el(Y) implies c 'imp' (a 'or' b)=I_el(Y) proof let a,b,c be Function of Y,BOOLEAN; assume A1: (c 'imp' a)=I_el(Y) & (c 'imp' b)=I_el(Y); c 'imp' (a 'or' b) =(c 'imp' a) 'or' (c 'imp' b) by Th73 .=I_el(Y) by A1; hence thesis; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'imp' c)=I_el(Y) & (b 'imp' c)=I_el(Y) implies (a '&' b) 'imp' c = I_el(Y) proof let a,b,c be Function of Y,BOOLEAN; (a 'imp' c)=I_el(Y) & (b 'imp' c)=I_el(Y) implies (a '&' b) 'imp' (c '&' c)=I_el(Y) by tt; hence thesis; end; theorem for a1,a2,b1,b2,c1,c2 being Function of Y,BOOLEAN holds (a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '<' (a2 'or' b2 'or' c2) proof let a1,a2,b1,b2,c1,c2 be Function of Y,BOOLEAN; let z be Element of Y; A1: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1)).z =((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2)).z '&' (a1 'or' b1 'or' c1).z by MARGREL1:def 20 .=((a1 'imp' a2) '&' (b1 'imp' b2)).z '&' (c1 'imp' c2).z '&' (a1 'or' b1 'or' c1).z by MARGREL1:def 20 .=(a1 'imp' a2).z '&' (b1 'imp' b2).z '&' (c1 'imp' c2).z '&' (a1 'or' b1 'or' c1).z by MARGREL1:def 20 .=('not' a1 'or' a2).z '&' (b1 'imp' b2).z '&' (c1 'imp' c2).z '&' (a1 'or' b1 'or' c1).z by BVFUNC_4:8 .=('not' a1 'or' a2).z '&' ('not' b1 'or' b2).z '&' (c1 'imp' c2).z '&' (a1 'or' b1 'or' c1).z by BVFUNC_4:8 .=('not' a1 'or' a2).z '&' ('not' b1 'or' b2).z '&' ('not' c1 'or' c2).z '&' (a1 'or' b1 'or' c1).z by BVFUNC_4:8 .=(('not' a1).z 'or' (a2).z) '&' ('not' b1 'or' b2).z '&' ('not' c1 'or' c2).z '&' (a1 'or' b1 'or' c1).z by BVFUNC_1:def 4 .=(('not' a1).z 'or' (a2).z) '&' (('not' b1).z 'or' (b2).z) '&' ('not' c1 'or' c2).z '&' (a1 'or' b1 'or' c1).z by BVFUNC_1:def 4 .=(('not' a1).z 'or' (a2).z) '&' (('not' b1).z 'or' (b2).z) '&' (('not' c1).z 'or' (c2).z) '&' (a1 'or' b1 'or' c1).z by BVFUNC_1:def 4 .=(('not' a1).z 'or' (a2).z) '&' (('not' b1).z 'or' (b2).z) '&' (('not' c1).z 'or' (c2).z) '&' ((a1 'or' b1).z 'or' (c1).z) by BVFUNC_1:def 4 .=(('not' a1).z 'or' (a2).z) '&' (('not' b1).z 'or' (b2).z) '&' (('not' c1).z 'or' (c2).z) '&' ((a1).z 'or' (b1).z 'or' (c1).z) by BVFUNC_1:def 4 .=('not' (a1).z 'or' (a2).z) '&' (('not' b1).z 'or' (b2).z) '&' (('not' c1).z 'or' (c2).z) '&' ((a1).z 'or' (b1).z 'or' (c1).z) by MARGREL1:def 19 .=('not' (a1).z 'or' (a2).z) '&' ('not' (b1).z 'or' (b2).z) '&' (('not' c1).z 'or' (c2).z) '&' ((a1).z 'or' (b1).z 'or' (c1).z) by MARGREL1:def 19 .=('not' (a1).z 'or' (a2).z) '&' ('not' (b1).z 'or' (b2).z) '&' ('not' ( c1).z 'or' (c2).z) '&' ((a1).z 'or' (b1).z 'or' (c1).z) by MARGREL1:def 19; assume A2: ((a1 'imp' a2) '&' (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1)).z=TRUE; now A3: b2.z=TRUE or b2.z=FALSE by XBOOLEAN:def 3; A4: c2.z=TRUE or c2.z=FALSE by XBOOLEAN:def 3; A5: ('not' a1 '&' 'not' b1 '&' 'not' c1) '&' (a1 'or' b1 'or' c1) =('not' a1 '&' 'not' b1 '&' 'not' c1) '&' (a1 'or' b1) 'or' ('not' a1 '&' 'not' b1 '&' 'not' c1) '&' c1 by BVFUNC_1:12 .=('not' a1 '&' 'not' b1 '&' 'not' c1) '&' (a1 'or' b1) 'or' ('not' a1 '&' 'not' b1 '&' ('not' c1 '&' c1)) by BVFUNC_1:4 .=('not' a1 '&' 'not' b1 '&' 'not' c1) '&' (a1 'or' b1) 'or' ('not' a1 '&' 'not' b1 '&' O_el(Y)) by BVFUNC_4:5 .=('not' a1 '&' 'not' b1 '&' 'not' c1) '&' (a1 'or' b1) 'or' O_el(Y) by BVFUNC_1:5 .=('not' a1 '&' 'not' b1 '&' 'not' c1) '&' (a1 'or' b1) by BVFUNC_1:9 .=('not' a1 '&' 'not' b1 '&' 'not' c1) '&' a1 'or' ('not' a1 '&' 'not' b1 '&' 'not' c1) '&' b1 by BVFUNC_1:12 .=('not' a1 '&' 'not' b1 '&' 'not' c1) '&' a1 'or' 'not' a1 '&' 'not' c1 '&' 'not' b1 '&' b1 by BVFUNC_1:4 .=('not' a1 '&' 'not' b1 '&' 'not' c1) '&' a1 'or' 'not' a1 '&' 'not' c1 '&' ('not' b1 '&' b1) by BVFUNC_1:4 .=('not' a1 '&' 'not' b1 '&' 'not' c1) '&' a1 'or' 'not' a1 '&' 'not' c1 '&' O_el(Y) by BVFUNC_4:5 .=('not' a1 '&' 'not' b1 '&' 'not' c1) '&' a1 'or' O_el(Y) by BVFUNC_1:5 .=('not' a1 '&' 'not' b1 '&' 'not' c1) '&' a1 by BVFUNC_1:9 .='not' b1 '&' 'not' c1 '&' 'not' a1 '&' a1 by BVFUNC_1:4 .='not' b1 '&' 'not' c1 '&' ('not' a1 '&' a1) by BVFUNC_1:4 .='not' b1 '&' 'not' c1 '&' O_el(Y) by BVFUNC_4:5 .=O_el(Y) by BVFUNC_1:5; A6: ((a2).z 'or' (b2).z)=TRUE or ((a2).z 'or' (b2).z)=FALSE by XBOOLEAN:def 3; A7: (a2 'or' b2 'or' c2).z =(a2 'or' b2).z 'or' (c2).z by BVFUNC_1:def 4 .=(a2).z 'or' b2.z 'or' c2.z by BVFUNC_1:def 4; assume (a2 'or' b2 'or' c2).z<>TRUE; then ('not' (a1).z 'or' (a2).z) '&' ('not' (b1).z 'or' (b2).z) '&' ('not' ( c1).z 'or' (c2).z) '&' ((a1).z 'or' (b1).z 'or' (c1).z) =(('not' a1).z '&' 'not' (b1).z '&' 'not' (c1).z) '&' ((a1).z 'or' (b1).z 'or' (c1).z) by A7,A6,A4 ,A3,MARGREL1:def 19 .=(('not' a1).z '&' ('not' b1).z '&' 'not' (c1).z) '&' ((a1).z 'or' ( b1).z 'or' (c1).z) by MARGREL1:def 19 .=(('not' a1).z '&' ('not' b1).z '&' ('not' c1).z) '&' ((a1).z 'or' ( b1).z 'or' (c1).z) by MARGREL1:def 19 .=(('not' a1).z '&' ('not' b1).z '&' ('not' c1).z) '&' ((a1 'or' b1).z 'or' (c1).z) by BVFUNC_1:def 4 .=(('not' a1).z '&' ('not' b1).z '&' ('not' c1).z) '&' ((a1 'or' b1 'or' c1).z) by BVFUNC_1:def 4 .=(('not' a1 '&' 'not' b1).z '&' ('not' c1).z) '&' ((a1 'or' b1 'or' c1).z) by MARGREL1:def 20 .=(('not' a1 '&' 'not' b1 '&' 'not' c1).z) '&' ((a1 'or' b1 'or' c1).z ) by MARGREL1:def 20 .=(('not' a1 '&' 'not' b1 '&' 'not' c1) '&' (a1 'or' b1 'or' c1)).z by MARGREL1:def 20; hence contradiction by A2,A1,A5,BVFUNC_1:def 10; end; hence thesis; end; Lm3: for a1,a2,b1,b2 being Function of Y,BOOLEAN holds (a1 'imp' b1) '&' (a2 'imp' b2) '&' (a1 'or' a2) '&' 'not'( b1 '&' b2) '<' (b1 'imp' a1) '&' (b2 'imp' a2) '&' (b1 'or' b2) '&' 'not'( a1 '&' a2) proof let a1,a2,b1,b2 be Function of Y,BOOLEAN; let z be Element of Y; A1: ((a1 'imp' b1) '&' (a2 'imp' b2) '&' (a1 'or' a2) '&' 'not'( b1 '&' b2)) .z =(('not' a1 'or' b1) '&' (a2 'imp' b2) '&' (a1 'or' a2) '&' 'not' (b1 '&' b2 )).z by BVFUNC_4:8 .=(('not' a1 'or' b1) '&' ('not' a2 'or' b2) '&' (a1 'or' a2) '&' 'not' (b1 '&' b2)).z by BVFUNC_4:8 .=(('not' a1 'or' b1) '&' ('not' a2 'or' b2) '&' (a1 'or' a2)).z '&' ( 'not'( b1 '&' b2)).z by MARGREL1:def 20 .=(('not' a1 'or' b1) '&' ('not' a2 'or' b2)).z '&' (a1 'or' a2).z '&' ( 'not'( b1 '&' b2)).z by MARGREL1:def 20 .=(('not' a1 'or' b1) '&' ('not' a2 'or' b2)).z '&' (a1 'or' a2).z '&' ( 'not' b1 'or' 'not' b2).z by BVFUNC_1:14 .=('not' a1 'or' b1).z '&' ('not' a2 'or' b2).z '&' (a1 'or' a2).z '&' ( 'not' b1 'or' 'not' b2).z by MARGREL1:def 20 .=(('not' a1).z 'or' (b1).z) '&' ('not' a2 'or' b2).z '&' (a1 'or' a2).z '&' ('not' b1 'or' 'not' b2).z by BVFUNC_1:def 4 .=(('not' a1).z 'or' (b1).z) '&' (('not' a2).z 'or' (b2).z) '&' (a1 'or' a2).z '&' ('not' b1 'or' 'not' b2).z by BVFUNC_1:def 4 .=(('not' a1).z 'or' (b1).z) '&' (('not' a2).z 'or' (b2).z) '&' ((a1).z 'or' (a2).z) '&' ('not' b1 'or' 'not' b2).z by BVFUNC_1:def 4 .=(('not' a1).z 'or' (b1).z) '&' (('not' a2).z 'or' (b2).z) '&' ((a1).z 'or' (a2).z) '&' (('not' b1).z 'or' ('not' b2).z) by BVFUNC_1:def 4 .=('not' (a1).z 'or' (b1).z) '&' (('not' a2).z 'or' (b2).z) '&' ((a1).z 'or' (a2).z) '&' (('not' b1).z 'or' ('not' b2).z) by MARGREL1:def 19 .=('not' (a1).z 'or' (b1).z) '&' ('not' (a2).z 'or' (b2).z) '&' ((a1).z 'or' (a2).z) '&' (('not' b1).z 'or' ('not' b2).z) by MARGREL1:def 19 .=('not' (a1).z 'or' (b1).z) '&' ('not' (a2).z 'or' (b2).z) '&' ((a1).z 'or' (a2).z) '&' ('not' (b1).z 'or' ('not' b2).z) by MARGREL1:def 19 .=('not' (a1).z 'or' (b1).z) '&' ('not' (a2).z 'or' (b2).z) '&' ((a1).z 'or' (a2).z) '&' ('not' (b1).z 'or' 'not' (b2).z) by MARGREL1:def 19; assume A2: ((a1 'imp' b1) '&' (a2 'imp' b2) '&' (a1 'or' a2) '&' 'not'( b1 '&' b2)).z=TRUE; now A3: ((b1 'imp' a1) '&' (b2 'imp' a2) '&' (b1 'or' b2) '&' 'not'( a1 '&' a2 )).z =(('not' b1 'or' a1) '&' (b2 'imp' a2) '&' (b1 'or' b2) '&' 'not' (a1 '&' a2)).z by BVFUNC_4:8 .=(('not' b1 'or' a1) '&' ('not' b2 'or' a2) '&' (b1 'or' b2) '&' 'not' (a1 '&' a2)).z by BVFUNC_4:8 .=(('not' b1 'or' a1) '&' ('not' b2 'or' a2) '&' (b1 'or' b2)).z '&' ( 'not'( a1 '&' a2)).z by MARGREL1:def 20 .=(('not' b1 'or' a1) '&' ('not' b2 'or' a2)).z '&' (b1 'or' b2).z '&' ('not'( a1 '&' a2)).z by MARGREL1:def 20 .=('not' b1 'or' a1).z '&' ('not' b2 'or' a2).z '&' (b1 'or' b2).z '&' ('not'( a1 '&' a2)).z by MARGREL1:def 20 .=('not' b1 'or' a1).z '&' ('not' b2 'or' a2).z '&' (b1 'or' b2).z '&' (('not' a1 'or' 'not' a2)).z by BVFUNC_1:14 .=(('not' b1).z 'or' (a1).z) '&' ('not' b2 'or' a2).z '&' (b1 'or' b2) .z '&' (('not' a1 'or' 'not' a2)).z by BVFUNC_1:def 4 .=(('not' b1).z 'or' (a1).z) '&' (('not' b2).z 'or' (a2).z) '&' (b1 'or' b2).z '&' (('not' a1 'or' 'not' a2)).z by BVFUNC_1:def 4 .=(('not' b1).z 'or' (a1).z) '&' (('not' b2).z 'or' (a2).z) '&' ((b1). z 'or' (b2).z) '&' (('not' a1 'or' 'not' a2)).z by BVFUNC_1:def 4 .=(('not' b1).z 'or' (a1).z) '&' (('not' b2).z 'or' (a2).z) '&' ((b1). z 'or' (b2).z) '&' (('not' a1).z 'or' ('not' a2).z) by BVFUNC_1:def 4; assume ((b1 'imp' a1) '&' (b2 'imp' a2) '&' (b1 'or' b2) '&' 'not'( a1 '&' a2)).z<>TRUE; then A4: (('not' b1).z 'or' (a1).z) '&' (('not' b2).z 'or' (a2).z) '&' ((b1).z 'or' (b2).z) '&' (('not' a1).z 'or' ('not' a2).z)=FALSE by A3,XBOOLEAN:def 3; now per cases by A4,MARGREL1:12; case A5: (('not' b1).z 'or' (a1).z) '&' (('not' b2).z 'or' (a2).z) '&' ((b1).z 'or' (b2).z)=FALSE; now per cases by A5,MARGREL1:12; case A6: (('not' b1).z 'or' (a1).z) '&' (('not' b2).z 'or' (a2).z) =FALSE; now per cases by A6,MARGREL1:12; case A7: (('not' b1).z 'or' (a1).z)=FALSE; A8: (a1).z=TRUE or (a1).z=FALSE by XBOOLEAN:def 3; then 'not' (b1).z=FALSE by A7,MARGREL1:def 19; then ('not' (a1).z 'or' (b1).z) '&' ('not' (a2).z 'or' (b2).z) '&' ((a1).z 'or' (a2).z) '&' ('not' (b1).z 'or' 'not' (b2).z) =((a2).z '&' 'not' (a2).z 'or' (a2).z '&' (b2).z) '&' 'not' (b2).z by A7,A8,XBOOLEAN:8 .=(FALSE 'or' (a2).z '&' (b2).z) '&' 'not' (b2).z by XBOOLEAN:138 .=((a2).z '&' (b2).z) '&' 'not' (b2).z .=(a2).z '&' ((b2).z '&' 'not' (b2).z) .=FALSE '&' (a2).z by XBOOLEAN:138 .=FALSE; hence thesis by A2,A1; end; case A9: (('not' b2).z 'or' (a2).z)=FALSE; A10: (a2).z=TRUE or (a2).z=FALSE by XBOOLEAN:def 3; then 'not' (b2).z=FALSE by A9,MARGREL1:def 19; then ('not' (a1).z 'or' (b1).z) '&' ('not' (a2).z 'or' (b2).z) '&' ((a1).z 'or' (a2).z) '&' ('not' (b1).z 'or' 'not' (b2).z) =((a1).z '&' 'not' (a1).z 'or' (a1).z '&' (b1).z) '&' 'not' (b1).z by A9,A10,XBOOLEAN:8 .=(FALSE 'or' (a1).z '&' (b1).z) '&' 'not' (b1).z by XBOOLEAN:138 .=((a1).z '&' (b1).z) '&' 'not' (b1).z .=(a1).z '&' ((b1).z '&' 'not' (b1).z) .=FALSE '&' (a1).z by XBOOLEAN:138 .=FALSE; hence thesis by A2,A1; end; end; hence thesis; end; case A11: b1.z 'or' b2.z=FALSE; reconsider a2z = a2.z as boolean object; reconsider a1z = a1.z as boolean object; (b1).z=TRUE or (b1).z=FALSE by XBOOLEAN:def 3; then ('not' (a1).z 'or' (b1).z) '&' ('not' (a2).z 'or' (b2).z) '&' ((a1).z 'or' (a2).z) '&' ('not' (b1).z 'or' 'not' (b2).z) ='not' (a1).z '&' ( 'not' (a2).z '&' ((a1).z 'or' (a2).z)) by A11 .='not' (a1).z '&' ('not' (a2).z '&' (a1).z 'or' 'not' a2z '&' a2z) by XBOOLEAN:8 .='not' (a1).z '&' ('not' (a2).z '&' (a1).z 'or' FALSE) by XBOOLEAN:138 .='not' (a1).z '&' ((a1).z '&' 'not' (a2).z) .='not' a1z '&' a1z '&' 'not' (a2).z .=FALSE '&' 'not' (a2).z by XBOOLEAN:138 .=FALSE; hence thesis by A2,A1; end; end; hence thesis; end; case A12: (('not' a1).z 'or' ('not' a2).z)=FALSE; ('not' a2).z=TRUE or ('not' a2).z=FALSE by XBOOLEAN:def 3; then 'not' (a1).z=FALSE & 'not' (a2).z=FALSE by A12,MARGREL1:def 19; then ('not' (a1).z 'or' (b1).z) '&' ('not' (a2).z 'or' (b2).z) '&' (( a1).z 'or' (a2).z) '&' ('not' (b1).z 'or' 'not' (b2).z) =(b1).z '&' ((b2).z '&' ('not' (b1).z 'or' 'not' (b2).z)) .=(b1).z '&' (b2.z '&' 'not' (b1).z 'or' (b2).z '&' 'not' (b2).z ) by XBOOLEAN:8 .=(b1).z '&' ((b2).z '&' 'not' (b1).z 'or' FALSE) by XBOOLEAN:138 .=(b1).z '&' ('not' (b1).z '&' (b2).z) .=(b1).z '&' 'not' (b1).z '&' (b2).z .=FALSE '&' (b2).z by XBOOLEAN:138 .=FALSE; hence thesis by A2,A1; end; end; hence thesis; end; hence thesis; end; theorem for a1,a2,b1,b2 being Function of Y,BOOLEAN holds (a1 'imp' b1) '&' (a2 'imp' b2) '&' (a1 'or' a2) '&' 'not'( b1 '&' b2)= (b1 'imp' a1) '&' (b2 'imp' a2) '&' (b1 'or' b2) '&' 'not'( a1 '&' a2) proof let a1,a2,b1,b2 be Function of Y,BOOLEAN; (a1 'imp' b1) '&' (a2 'imp' b2) '&' (a1 'or' a2) '&' 'not'( b1 '&' b2) '<' ( b1 'imp' a1) '&' (b2 'imp' a2) '&' (b1 'or' b2) '&' 'not'( a1 '&' a2) & ( b1 'imp' a1) '&' (b2 'imp' a2) '&' (b1 'or' b2) '&' 'not'( a1 '&' a2) '<' (a1 'imp' b1) '&' (a2 'imp' b2) '&' (a1 'or' a2) '&' 'not'( b1 '&' b2) by Lm3; hence thesis by BVFUNC_1:15; end; theorem for a,b,c,d being Function of Y,BOOLEAN holds (a 'or' b) '&' (c 'or' d) = (a '&' c) 'or' (a '&' d) 'or' (b '&' c) 'or' (b '&' d) proof let a,b,c,d be Function of Y,BOOLEAN; (a 'or' b) '&' (c 'or' d) =((a 'or' b) '&' c) 'or' ((a 'or' b) '&' d) by BVFUNC_1:12 .=(a '&' c) 'or' (b '&' c) 'or' ((a 'or' b) '&' d) by BVFUNC_1:12 .=(a '&' c) 'or' (b '&' c) 'or' ((a '&' d) 'or' (b '&' d)) by BVFUNC_1:12 .=(a '&' c) 'or' (b '&' c) 'or' (a '&' d) 'or' (b '&' d) by BVFUNC_1:8 .=(a '&' c) 'or' (a '&' d) 'or' (b '&' c) 'or' (b '&' d) by BVFUNC_1:8; hence thesis; end; theorem for a1,a2,b1,b2,b3 being Function of Y,BOOLEAN holds (a1 '&' a2) 'or' (b1 '&' b2 '&' b3)= (a1 'or' b1) '&' (a1 'or' b2) '&' (a1 'or' b3) '&' (a2 'or' b1) '&' (a2 'or' b2) '&' (a2 'or' b3) proof let a1,a2,b1,b2,b3 be Function of Y,BOOLEAN; (a1 'or' b1) '&' (a1 'or' b2) '&' (a1 'or' b3) '&' (a2 'or' b1) '&' (a2 'or' b2) '&' (a2 'or' b3) =(a1 'or' (b1 '&' b2)) '&' (a1 'or' b3) '&' (a2 'or' b1) '&' (a2 'or' b2) '&' (a2 'or' b3) by BVFUNC_1:11 .=(a1 'or' (b1 '&' b2 '&' b3)) '&' (a2 'or' b1) '&' (a2 'or' b2) '&' (a2 'or' b3) by BVFUNC_1:11 .=(a1 'or' (b1 '&' b2 '&' b3)) '&' ((a2 'or' b1) '&' (a2 'or' b2)) '&' ( a2 'or' b3) by BVFUNC_1:4 .=(a1 'or' (b1 '&' b2 '&' b3)) '&' (((a2 'or' b1) '&' (a2 'or' b2)) '&' (a2 'or' b3)) by BVFUNC_1:4 .=(a1 'or' (b1 '&' b2 '&' b3)) '&' ((a2 'or' (b1 '&' b2)) '&' (a2 'or' b3)) by BVFUNC_1:11 .=(a1 'or' (b1 '&' b2 '&' b3)) '&' (a2 'or' (b1 '&' b2 '&' b3)) by BVFUNC_1:11 .=(a1 '&' a2) 'or' (b1 '&' b2 '&' b3) by BVFUNC_1:11; hence thesis; end; theorem for a,b,c,d being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' d)= (a 'imp' (b '&' c '&' d)) '&' (b 'imp' (c '&' d)) '&' (c 'imp' d) proof let a,b,c,d be Function of Y,BOOLEAN; A1: (a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' d) =(a 'imp' b) '&' (b 'imp' c) '&' (a 'imp' c) '&' (c 'imp' d) by Th12 .=(a 'imp' b) '&' (b 'imp' c) '&' ((a 'imp' c) '&' (c 'imp' d)) by BVFUNC_1:4 .=(a 'imp' b) '&' (b 'imp' c) '&' ((a 'imp' c) '&' (c 'imp' d) '&' (a 'imp' d)) by Th12 .=(a 'imp' b) '&' (b 'imp' c) '&' ((a 'imp' c) '&' (c 'imp' d)) '&' (a 'imp' d) by BVFUNC_1:4 .=(a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' d) '&' (a 'imp' c) '&' (a 'imp' d) by BVFUNC_1:4 .=(a 'imp' b) '&' ((b 'imp' c) '&' (c 'imp' d)) '&' (a 'imp' c) '&' (a 'imp' d) by BVFUNC_1:4 .=(a 'imp' b) '&' (((b 'imp' c) '&' (c 'imp' d)) '&' (b 'imp' d)) '&' (a 'imp' c) '&' (a 'imp' d) by Th12 .=(a 'imp' b) '&' ((b 'imp' c) '&' (c 'imp' d)) '&' (b 'imp' d) '&' (a 'imp' c) '&' (a 'imp' d) by BVFUNC_1:4 .=(a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' d) '&' (b 'imp' d) '&' (a 'imp' c) '&' (a 'imp' d) by BVFUNC_1:4 .=(a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' d) '&' (a 'imp' c) '&' (b 'imp' d) '&' (a 'imp' d) by BVFUNC_1:4 .=(a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' d) '&' (a 'imp' c) '&' (a 'imp' d) '&' (b 'imp' d) by BVFUNC_1:4; (a 'imp' (b '&' c '&' d)) '&' (b 'imp' (c '&' d)) '&' (c 'imp' d) =( 'not' a 'or' (b '&' c '&' d)) '&' (b 'imp' (c '&' d)) '&' (c 'imp' d) by BVFUNC_4:8 .=('not' a 'or' (b '&' c '&' d)) '&' ('not' b 'or' (c '&' d)) '&' (c 'imp' d) by BVFUNC_4:8 .=('not' a 'or' (b '&' c '&' d)) '&' ('not' b 'or' (c '&' d)) '&' ('not' c 'or' d) by BVFUNC_4:8 .=(('not' a 'or' b) '&' ('not' a 'or' c) '&' ('not' a 'or' d)) '&' ( 'not' b 'or' (c '&' d)) '&' ('not' c 'or' d) by BVFUNC_5:39 .=(('not' a 'or' b) '&' ('not' a 'or' c) '&' ('not' a 'or' d)) '&' (( 'not' b 'or' c) '&' ('not' b 'or' d)) '&' ('not' c 'or' d) by BVFUNC_1:11 .=('not' a 'or' b) '&' ('not' a 'or' c) '&' ('not' a 'or' d) '&' ('not' b 'or' c) '&' ('not' b 'or' d) '&' ('not' c 'or' d) by BVFUNC_1:4 .=('not' a 'or' b) '&' ('not' a 'or' c) '&' ('not' b 'or' c) '&' ('not' a 'or' d) '&' ('not' b 'or' d) '&' ('not' c 'or' d) by BVFUNC_1:4 .=('not' a 'or' b) '&' ('not' b 'or' c) '&' ('not' a 'or' c) '&' ('not' a 'or' d) '&' ('not' b 'or' d) '&' ('not' c 'or' d) by BVFUNC_1:4 .=('not' a 'or' b) '&' ('not' b 'or' c) '&' ('not' a 'or' c) '&' ('not' a 'or' d) '&' ('not' c 'or' d) '&' ('not' b 'or' d) by BVFUNC_1:4 .=('not' a 'or' b) '&' ('not' b 'or' c) '&' ('not' a 'or' c) '&' ('not' c 'or' d) '&' ('not' a 'or' d) '&' ('not' b 'or' d) by BVFUNC_1:4 .=('not' a 'or' b) '&' ('not' b 'or' c) '&' ('not' c 'or' d) '&' ('not' a 'or' c) '&' ('not' a 'or' d) '&' ('not' b 'or' d) by BVFUNC_1:4 .=(a 'imp' b) '&' ('not' b 'or' c) '&' ('not' c 'or' d) '&' ('not' a 'or' c) '&' ('not' a 'or' d) '&' ('not' b 'or' d) by BVFUNC_4:8 .=(a 'imp' b) '&' (b 'imp' c) '&' ('not' c 'or' d) '&' ('not' a 'or' c) '&' ('not' a 'or' d) '&' ('not' b 'or' d) by BVFUNC_4:8 .=(a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' d) '&' ('not' a 'or' c) '&' ( 'not' a 'or' d) '&' ('not' b 'or' d) by BVFUNC_4:8 .=(a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' d) '&' (a 'imp' c) '&' ('not' a 'or' d) '&' ('not' b 'or' d) by BVFUNC_4:8 .=(a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' d) '&' (a 'imp' c) '&' (a 'imp' d) '&' ('not' b 'or' d) by BVFUNC_4:8 .=(a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' d) '&' (a 'imp' c) '&' (a 'imp' d) '&' (b 'imp' d) by BVFUNC_4:8; hence thesis by A1; end; theorem for a,b,c,d being Function of Y,BOOLEAN holds (a 'imp' c) '&' (b 'imp' d) '&' (a 'or' b) '<' (c 'or' d) proof let a,b,c,d be Function of Y,BOOLEAN; let z be Element of Y; A1: ((a 'imp' c) '&' (b 'imp' d) '&' (a 'or' b)).z =((a 'imp' c) '&' (b 'imp' d)).z '&' (a 'or' b).z by MARGREL1:def 20 .=(a 'imp' c).z '&' (b 'imp' d).z '&' (a 'or' b).z by MARGREL1:def 20 .=('not' a 'or' c).z '&' (b 'imp' d).z '&' (a 'or' b).z by BVFUNC_4:8 .=('not' a 'or' c).z '&' ('not' b 'or' d).z '&' (a 'or' b).z by BVFUNC_4:8 .=(('not' a).z 'or' c.z) '&' ('not' b 'or' d).z '&' (a 'or' b).z by BVFUNC_1:def 4 .=(('not' a).z 'or' c.z) '&' (('not' b).z 'or' (d).z) '&' (a 'or' b).z by BVFUNC_1:def 4 .=(('not' a).z 'or' c.z) '&' (('not' b).z 'or' (d).z) '&' (a.z 'or' b.z) by BVFUNC_1:def 4 .=('not' a.z 'or' c.z) '&' (('not' b).z 'or' (d).z) '&' (a.z 'or' b.z) by MARGREL1:def 19 .=('not' a.z 'or' c.z) '&' ('not' b.z 'or' (d).z) '&' (a.z 'or' b.z) by MARGREL1:def 19; reconsider bz = b.z as boolean object; reconsider az = a.z as boolean object; assume A2: ((a 'imp' c) '&' (b 'imp' d) '&' (a 'or' b)).z=TRUE; now assume (c 'or' d).z<>TRUE; then (c 'or' d).z=FALSE by XBOOLEAN:def 3; then A3: c.z 'or' (d).z=FALSE by BVFUNC_1:def 4; (d).z=TRUE or (d).z=FALSE by XBOOLEAN:def 3; then ('not' a.z 'or' c.z) '&' ('not' b.z 'or' (d).z) '&' (a.z 'or' b.z) ='not' a.z '&' ('not' b.z '&' (b.z 'or' a.z)) by A3 .='not' a.z '&' ( ('not' bz '&' bz 'or' 'not' b.z '&' a.z)) by XBOOLEAN:8 .='not' a.z '&' ( (FALSE 'or' 'not' b.z '&' a.z)) by XBOOLEAN:138 .='not' a.z '&' (a.z '&' 'not' b.z) .='not' az '&' az '&' 'not' b.z .=FALSE '&' 'not' b.z by XBOOLEAN:138 .=FALSE; hence thesis by A2,A1; end; hence thesis; end; theorem for a,b,c being Function of Y,BOOLEAN holds ((a '&' b) 'imp' 'not' c) '&' a '&' c '<' 'not' b proof let a,b,c be Function of Y,BOOLEAN; let z be Element of Y; A1: (((a '&' b) 'imp' 'not' c) '&' a '&' c).z =(((a '&' b) 'imp' 'not' c) '&' a).z '&' c.z by MARGREL1:def 20 .=((a '&' b) 'imp' 'not' c).z '&' a.z '&' c.z by MARGREL1:def 20 .=('not'( a '&' b) 'or' 'not' c).z '&' a.z '&' c.z by BVFUNC_4:8 .=(('not' a 'or' 'not' b) 'or' 'not' c).z '&' a.z '&' c.z by BVFUNC_1:14 .=(('not' a 'or' 'not' b).z 'or' ('not' c).z) '&' a.z '&' c.z by BVFUNC_1:def 4 .=((('not' a).z 'or' ('not' b).z) 'or' ('not' c).z) '&' a.z '&' c.z by BVFUNC_1:def 4; reconsider cz = c.z as boolean object; assume A2: (((a '&' b) 'imp' 'not' c) '&' a '&' c).z=TRUE; now assume ('not' b).z<>TRUE; then ('not' b).z=FALSE by XBOOLEAN:def 3; then ((('not' a).z 'or' ('not' b).z) 'or' ('not' c).z) '&' a.z '&' c.z =('not' a.z 'or' ('not' c).z) '&' a.z '&' c.z by MARGREL1:def 19 .=a.z '&' ('not' a.z 'or' 'not' c.z) '&' c.z by MARGREL1:def 19 .=(a.z '&' 'not' a.z 'or' a.z '&' 'not' c.z) '&' c.z by XBOOLEAN:8 .=(FALSE 'or' a.z '&' 'not' c.z) '&' c.z by XBOOLEAN:138 .=(a.z '&' 'not' c.z) '&' c.z .=a.z '&' ('not' cz '&' cz) .=FALSE '&' a.z by XBOOLEAN:138 .=FALSE; hence thesis by A2,A1; end; hence thesis; end; theorem for a1,a2,a3,b1,b2,b3 being Function of Y,BOOLEAN holds (a1 '&' a2 '&' a3) 'imp' (b1 'or' b2 'or' b3)= ('not' b1 '&' 'not' b2 '&' a3) 'imp' ( 'not' a1 'or' 'not' a2 'or' b3) proof let a1,a2,a3,b1,b2,b3 be Function of Y,BOOLEAN; ('not' b1 '&' 'not' b2 '&' a3) 'imp' ('not' a1 'or' 'not' a2 'or' b3) = 'not'( 'not' b1 '&' 'not' b2 '&' a3) 'or' ('not' a1 'or' 'not' a2 'or' b3) by BVFUNC_4:8 .=('not' 'not' b1 'or' 'not' 'not' b2 'or' 'not' a3) 'or' ('not' a1 'or' 'not' a2 'or' b3) by BVFUNC_5:37 .=b1 'or' b2 'or' 'not' a3 'or' ('not' a1 'or' 'not' a2) 'or' b3 by BVFUNC_1:8 .=b1 'or' b2 'or' (('not' a1 'or' 'not' a2) 'or' 'not' a3) 'or' b3 by BVFUNC_1:8 .=(('not' a1 'or' 'not' a2) 'or' 'not' a3) 'or' ((b1 'or' b2) 'or' b3) by BVFUNC_1:8 .='not'( a1 '&' a2 '&' a3) 'or' (b1 'or' b2 'or' b3) by BVFUNC_5:37 .=(a1 '&' a2 '&' a3) 'imp' (b1 'or' b2 'or' b3) by BVFUNC_4:8; hence thesis; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a) = (a '&' b '&' c) 'or' ('not' a '&' 'not' b '&' 'not' c) proof let a,b,c be Function of Y,BOOLEAN; for z being Element of Y st ((a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a) ).z=TRUE holds ((a '&' b '&' c) 'or' ('not' a '&' 'not' b '&' 'not' c)).z=TRUE proof let z be Element of Y; A1: ((a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a)).z =((a 'imp' b) '&' (b 'imp' c)).z '&' (c 'imp' a).z by MARGREL1:def 20 .=(a 'imp' b).z '&' (b 'imp' c).z '&' (c 'imp' a).z by MARGREL1:def 20 .=('not' a 'or' b).z '&' (b 'imp' c).z '&' (c 'imp' a).z by BVFUNC_4:8 .=('not' a 'or' b).z '&' ('not' b 'or' c).z '&' (c 'imp' a).z by BVFUNC_4:8 .=('not' a 'or' b).z '&' ('not' b 'or' c).z '&' ('not' c 'or' a).z by BVFUNC_4:8 .=(('not' a).z 'or' b.z) '&' ('not' b 'or' c).z '&' ('not' c 'or' a) .z by BVFUNC_1:def 4 .=(('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) '&' ('not' c 'or' a).z by BVFUNC_1:def 4 .=(('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) '&' (('not' c) .z 'or' a.z) by BVFUNC_1:def 4 .=('not' a.z 'or' b.z) '&' (('not' b).z 'or' c.z) '&' (('not' c) .z 'or' a.z) by MARGREL1:def 19 .=('not' a.z 'or' b.z) '&' ('not' b.z 'or' c.z) '&' (('not' c) .z 'or' a.z) by MARGREL1:def 19 .=('not' a.z 'or' b.z) '&' ('not' b.z 'or' c.z) '&' ('not' (c) .z 'or' a.z) by MARGREL1:def 19; assume A2: ((a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a)).z=TRUE; now A3: ('not' a.z '&' 'not' b.z '&' 'not' c.z)=TRUE or ('not' a.z '&' 'not' b.z '&' 'not' c.z)=FALSE by XBOOLEAN:def 3; assume A4: ((a '&' b '&' c) 'or' ('not' a '&' 'not' b '&' 'not' c)).z<>TRUE; A5: ((a '&' b '&' c) 'or' ('not' a '&' 'not' b '&' 'not' c)).z =(a '&' b '&' c).z 'or' ('not' a '&' 'not' b '&' 'not' c).z by BVFUNC_1:def 4 .=((a '&' b).z '&' c.z) 'or' ('not' a '&' 'not' b '&' 'not' c).z by MARGREL1:def 20 .=(a.z '&' b.z '&' c.z) 'or' ('not' a '&' 'not' b '&' 'not' c) .z by MARGREL1:def 20 .=(a.z '&' b.z '&' c.z) 'or' (('not' a '&' 'not' b).z '&' ( 'not' c).z) by MARGREL1:def 20 .=(a.z '&' b.z '&' c.z) 'or' (('not' a).z '&' ('not' b).z '&' ('not' c).z) by MARGREL1:def 20 .=(a.z '&' b.z '&' c.z) 'or' ('not' a.z '&' ('not' b).z '&' ('not' c).z) by MARGREL1:def 19 .=(a.z '&' b.z '&' c.z) 'or' ('not' a.z '&' 'not' b.z '&' ('not' c).z) by MARGREL1:def 19 .=(a.z '&' b.z '&' c.z) 'or' ('not' a.z '&' 'not' b.z '&' 'not' c.z) by MARGREL1:def 19; A6: (a.z '&' b.z '&' c.z)=TRUE or (a.z '&' b.z '&' c.z)= FALSE by XBOOLEAN:def 3; now per cases by A4,A5,A6,MARGREL1:12; case A7: (a.z '&' b.z)=FALSE; now per cases by A7,MARGREL1:12; case A8: a.z=FALSE; now per cases by A4,A5,A3,A8,MARGREL1:12; case 'not' b.z=FALSE; hence thesis by A2,A1,A8,XBOOLEAN:138; end; case 'not' c.z=FALSE; hence thesis by A2,A1,A8; end; end; hence thesis; end; case A9: b.z=FALSE; now per cases by A4,A5,A3,A9,MARGREL1:12; case 'not' a.z=FALSE; hence thesis by A2,A1,A9; end; case 'not' c.z=FALSE; hence thesis by A2,A1,A9,XBOOLEAN:138; end; end; hence thesis; end; end; hence thesis; end; case A10: c.z=FALSE; now per cases by A4,A5,A3,A10,MARGREL1:12; case 'not' a.z=FALSE; hence thesis by A2,A1,A10,XBOOLEAN:138; end; case 'not' b.z=FALSE; hence thesis by A2,A1,A10; end; end; hence thesis; end; end; hence thesis; end; hence thesis; end; then A11: (a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a) '<' (a '&' b '&' c) 'or' ( 'not' a '&' 'not' b '&' 'not' c); for z being Element of Y st ((a '&' b '&' c) 'or' ('not' a '&' 'not' b '&' 'not' c)).z=TRUE holds ((a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a)).z=TRUE proof let z be Element of Y; A12: ((a '&' b '&' c) 'or' ('not' a '&' 'not' b '&' 'not' c)).z =(a '&' b '&' c).z 'or' ('not' a '&' 'not' b '&' 'not' c).z by BVFUNC_1:def 4 .=((a '&' b).z '&' c.z) 'or' ('not' a '&' 'not' b '&' 'not' c).z by MARGREL1:def 20 .=(a.z '&' b.z '&' c.z) 'or' ('not' a '&' 'not' b '&' 'not' c).z by MARGREL1:def 20 .=(a.z '&' b.z '&' c.z) 'or' (('not' a '&' 'not' b).z '&' ('not' c).z) by MARGREL1:def 20 .=(a.z '&' b.z '&' c.z) 'or' (('not' a).z '&' ('not' b).z '&' ( 'not' c).z) by MARGREL1:def 20 .=(a.z '&' b.z '&' c.z) 'or' ('not' a.z '&' ('not' b).z '&' ( 'not' c).z) by MARGREL1:def 19 .=(a.z '&' b.z '&' c.z) 'or' ('not' a.z '&' 'not' b.z '&' ( 'not' c).z) by MARGREL1:def 19 .=(a.z '&' b.z '&' c.z) 'or' ('not' a.z '&' 'not' b.z '&' 'not' c.z) by MARGREL1:def 19; assume A13: ((a '&' b '&' c) 'or' ('not' a '&' 'not' b '&' 'not' c)).z=TRUE; now A14: ((a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a)).z =((a 'imp' b) '&' ( b 'imp' c)).z '&' (c 'imp' a).z by MARGREL1:def 20 .=(a 'imp' b).z '&' (b 'imp' c).z '&' (c 'imp' a).z by MARGREL1:def 20 .=('not' a 'or' b).z '&' (b 'imp' c).z '&' (c 'imp' a).z by BVFUNC_4:8 .=('not' a 'or' b).z '&' ('not' b 'or' c).z '&' (c 'imp' a).z by BVFUNC_4:8 .=('not' a 'or' b).z '&' ('not' b 'or' c).z '&' ('not' c 'or' a).z by BVFUNC_4:8 .=(('not' a).z 'or' b.z) '&' ('not' b 'or' c).z '&' ('not' c 'or' a).z by BVFUNC_1:def 4 .=(('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) '&' ('not' c 'or' a).z by BVFUNC_1:def 4 .=(('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) '&' (('not' c).z 'or' a.z) by BVFUNC_1:def 4 .=('not' a.z 'or' b.z) '&' (('not' b).z 'or' c.z) '&' (('not' c).z 'or' a.z) by MARGREL1:def 19 .=('not' a.z 'or' b.z) '&' ('not' b.z 'or' c.z) '&' (('not' c).z 'or' a.z) by MARGREL1:def 19 .=('not' a.z 'or' b.z) '&' ('not' b.z 'or' c.z) '&' ('not' ( c).z 'or' a.z) by MARGREL1:def 19; assume ((a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a)).z<>TRUE; then A15: ('not' a.z 'or' b.z) '&' ('not' b.z 'or' c.z) '&' ('not' (c ).z 'or' a.z)=FALSE by A14,XBOOLEAN:def 3; now per cases by A15,MARGREL1:12; case A16: ('not' a.z 'or' b.z) '&' ('not' b.z 'or' c.z)= FALSE; now per cases by A16,MARGREL1:12; case A17: ('not' a.z 'or' b.z)=FALSE; b.z=TRUE or b.z=FALSE by XBOOLEAN:def 3; hence thesis by A13,A12,A17; end; case A18: ('not' b.z 'or' c.z)=FALSE; c.z=TRUE or c.z=FALSE by XBOOLEAN:def 3; hence thesis by A13,A12,A18; end; end; hence thesis; end; case A19: ('not' c.z 'or' a.z)=FALSE; a.z=TRUE or a.z=FALSE by XBOOLEAN:def 3; hence thesis by A13,A12,A19; end; end; hence thesis; end; hence thesis; end; then (a '&' b '&' c) 'or' ('not' a '&' 'not' b '&' 'not' c) '<' (a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a); hence thesis by A11,BVFUNC_1:15; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a) '&' (a 'or' b 'or' c)= (a '&' b '&' c) proof let a,b,c be Function of Y,BOOLEAN; (a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a) '&' (a 'or' b 'or' c) =( 'not' a 'or' b) '&' (b 'imp' c) '&' (c 'imp' a) '&' (a 'or' b 'or' c) by BVFUNC_4:8 .=('not' a 'or' b) '&' ('not' b 'or' c) '&' (c 'imp' a) '&' (a 'or' b 'or' c) by BVFUNC_4:8 .=('not' a 'or' b) '&' ('not' b 'or' c) '&' ('not' c 'or' a) '&' (a 'or' b 'or' c) by BVFUNC_4:8 .=('not' a 'or' b) '&' ('not' b 'or' c) '&' (('not' c 'or' a) '&' (a 'or' b 'or' c)) by BVFUNC_1:4 .=('not' a 'or' b) '&' ('not' b 'or' c) '&' (('not' c 'or' a) '&' (a 'or' b) 'or' ('not' c 'or' a) '&' c) by BVFUNC_1:12 .=('not' a 'or' b) '&' ('not' b 'or' c) '&' (('not' c 'or' a) '&' (a 'or' b) 'or' ('not' c '&' c 'or' a '&' c)) by BVFUNC_1:12 .=('not' a 'or' b) '&' ('not' b 'or' c) '&' (('not' c 'or' a) '&' (a 'or' b) 'or' (O_el(Y) 'or' a '&' c)) by BVFUNC_4:5 .=('not' a 'or' b) '&' ('not' b 'or' c) '&' (('not' c 'or' a) '&' (a 'or' b) 'or' (a '&' c)) by BVFUNC_1:9 .=('not' a 'or' b) '&' ('not' b 'or' c) '&' (a 'or' ('not' c '&' b) 'or' (a '&' c)) by BVFUNC_1:11 .=('not' a 'or' b) '&' ('not' b 'or' c) '&' (a 'or' (a '&' c) 'or' ( 'not' c '&' b)) by BVFUNC_1:8 .=('not' a 'or' b) '&' ('not' b 'or' c) '&' ((a '&' I_el(Y)) 'or' (a '&' c) 'or' ('not' c '&' b)) by BVFUNC_1:6 .=('not' a 'or' b) '&' ('not' b 'or' c) '&' ((a '&' (I_el(Y) 'or' c)) 'or' ('not' c '&' b)) by BVFUNC_1:12 .=('not' a 'or' b) '&' ('not' b 'or' c) '&' ((a '&' I_el(Y)) 'or' ('not' c '&' b)) by BVFUNC_1:10 .=('not' a 'or' b) '&' ('not' b 'or' c) '&' (a 'or' ('not' c '&' b)) by BVFUNC_1:6 .=('not' a 'or' b) '&' ('not' b 'or' c) '&' ((a 'or' 'not' c) '&' (a 'or' b)) by BVFUNC_1:11 .=(a 'or' b) '&' (('not' a 'or' b) '&' ('not' b 'or' c)) '&' (a 'or' 'not' c) by BVFUNC_1:4 .=(a 'or' b) '&' ('not' a 'or' b) '&' ('not' b 'or' c) '&' (a 'or' 'not' c) by BVFUNC_1:4 .=((a '&' 'not' a) 'or' b) '&' ('not' b 'or' c) '&' (a 'or' 'not' c) by BVFUNC_1:11 .=(O_el(Y) 'or' b) '&' ('not' b 'or' c) '&' (a 'or' 'not' c) by BVFUNC_4:5 .=b '&' ('not' b 'or' c) '&' (a 'or' 'not' c) by BVFUNC_1:9 .=(b '&' 'not' b 'or' b '&' c) '&' (a 'or' 'not' c) by BVFUNC_1:12 .=(O_el(Y) 'or' b '&' c) '&' (a 'or' 'not' c) by BVFUNC_4:5 .=(b '&' c) '&' (a 'or' 'not' c) by BVFUNC_1:9 .=(b '&' c) '&' a 'or' (b '&' c) '&' 'not' c by BVFUNC_1:12 .=(b '&' c) '&' a 'or' b '&' (c '&' 'not' c) by BVFUNC_1:4 .=(b '&' c) '&' a 'or' b '&' O_el(Y) by BVFUNC_4:5 .=(b '&' c) '&' a 'or' O_el(Y) by BVFUNC_1:5 .=(b '&' c) '&' a by BVFUNC_1:9 .=(a '&' b '&' c) by BVFUNC_1:4; hence thesis; end; Lm4: for a,b,c being Function of Y,BOOLEAN holds ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c) '<' (a 'or' b) '&' 'not'( a '&' b '&' c) proof let a,b,c be Function of Y,BOOLEAN; let z be Element of Y; A1: (('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c)).z =(('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c)).z 'or' (a '&' b '&' 'not' c).z by BVFUNC_1:def 4 .=('not' a '&' b '&' c).z 'or' (a '&' 'not' b '&' c).z 'or' (a '&' b '&' 'not' c).z by BVFUNC_1:def 4 .=('not' a '&' b '&' c).z 'or' (a '&' 'not' b '&' c).z 'or' ((a '&' b).z '&' ('not' c).z) by MARGREL1:def 20 .=('not' a '&' b '&' c).z 'or' (a '&' 'not' b '&' c).z 'or' (a.z '&' ( b).z '&' ('not' c).z) by MARGREL1:def 20 .=('not' a '&' b '&' c).z 'or' ((a '&' 'not' b).z '&' c.z) 'or' (a.z '&' b.z '&' ('not' c).z) by MARGREL1:def 20 .=('not' a '&' b '&' c).z 'or' (a.z '&' ('not' b).z '&' c.z) 'or' (( a).z '&' b.z '&' ('not' c).z) by MARGREL1:def 20 .=(('not' a '&' b).z '&' c.z) 'or' (a.z '&' ('not' b).z '&' c.z) 'or' (a.z '&' b.z '&' ('not' c).z) by MARGREL1:def 20 .=(('not' a).z '&' b.z '&' c.z) 'or' (a.z '&' ('not' b).z '&' c. z) 'or' (a.z '&' b.z '&' ('not' c).z) by MARGREL1:def 20 .=('not' a.z '&' b.z '&' c.z) 'or' (a.z '&' ('not' b).z '&' c. z) 'or' (a.z '&' b.z '&' ('not' c).z) by MARGREL1:def 19 .=('not' a.z '&' b.z '&' c.z) 'or' (a.z '&' 'not' b.z '&' c. z) 'or' (a.z '&' b.z '&' ('not' c).z) by MARGREL1:def 19 .=('not' a.z '&' b.z '&' c.z) 'or' (a.z '&' 'not' b.z '&' c. z) 'or' (a.z '&' b.z '&' 'not' c.z) by MARGREL1:def 19; assume A2: (('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c)).z=TRUE; now A3: ((a 'or' b) '&' 'not'( a '&' b '&' c)).z =((a 'or' b) '&' ('not' a 'or' 'not' b 'or' 'not' c)).z by BVFUNC_5:37 .=(a 'or' b).z '&' ('not' a 'or' 'not' b 'or' 'not' c).z by MARGREL1:def 20 .=(a.z 'or' b.z) '&' ('not' a 'or' 'not' b 'or' 'not' c).z by BVFUNC_1:def 4 .=(a.z 'or' b.z) '&' (('not' a 'or' 'not' b).z 'or' ('not' c).z) by BVFUNC_1:def 4 .=(a.z 'or' b.z) '&' (('not' a).z 'or' ('not' b).z 'or' ('not' c). z) by BVFUNC_1:def 4 .=(a.z 'or' b.z) '&' ('not' a.z 'or' ('not' b).z 'or' ('not' c). z) by MARGREL1:def 19 .=(a.z 'or' b.z) '&' ('not' a.z 'or' 'not' b.z 'or' ('not' c). z) by MARGREL1:def 19 .=(a.z 'or' b.z) '&' ('not' a.z 'or' 'not' b.z 'or' 'not' c. z) by MARGREL1:def 19; assume ((a 'or' b) '&' 'not'( a '&' b '&' c)).z<>TRUE; then A4: (a.z 'or' b.z) '&' ('not' a.z 'or' 'not' b.z 'or' 'not' c.z) =FALSE by A3,XBOOLEAN:def 3; now per cases by A4,MARGREL1:12; case A5: (a.z 'or' b.z)=FALSE; b.z=TRUE or b.z=FALSE by XBOOLEAN:def 3; hence thesis by A2,A1,A5; end; case A6: ('not' a.z 'or' 'not' b.z 'or' 'not' c.z)=FALSE; A7: 'not' b.z=TRUE or 'not' b.z=FALSE by XBOOLEAN:def 3; 'not' a.z 'or' 'not' b.z=TRUE or 'not' a.z 'or' 'not' b.z =FALSE by XBOOLEAN:def 3; hence thesis by A2,A1,A6,A7; end; end; hence thesis; end; hence thesis; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'or' b) '&' (b 'or' c) '&' (c 'or' a) '&' 'not'( a '&' b '&' c)= ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c) proof let a,b,c be Function of Y,BOOLEAN; A1: (a 'or' b) '&' (b 'or' c) '&' 'not'( a '&' b '&' c) '&' ((c 'or' a) '&' 'not'( a '&' b '&' c)) =(a 'or' b) '&' (b 'or' c) '&' 'not'( a '&' b '&' c) '&' (c 'or' a) '&' 'not'( a '&' b '&' c) by BVFUNC_1:4 .=(a 'or' b) '&' (b 'or' c) '&' (c 'or' a) '&' 'not'( a '&' b '&' c) '&' 'not'( a '&' b '&' c) by BVFUNC_1:4 .=(a 'or' b) '&' (b 'or' c) '&' (c 'or' a) '&' ('not'( a '&' b '&' c) '&' 'not'( a '&' b '&' c)) by BVFUNC_1:4 .=(a 'or' b) '&' (b 'or' c) '&' (c 'or' a) '&' 'not'( a '&' b '&' c); for z being Element of Y st ((a 'or' b) '&' (b 'or' c) '&' (c 'or' a) '&' 'not' (a '&' b '&' c)).z=TRUE holds (('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c)).z=TRUE proof let z be Element of Y; A2: ((a 'or' b) '&' (b 'or' c) '&' (c 'or' a) '&' 'not'( a '&' b '&' c)).z =((a 'or' b) '&' (b 'or' c) '&' (c 'or' a)).z '&' ('not' (a '&' b '&' c)).z by MARGREL1:def 20 .=((a 'or' b) '&' (b 'or' c)).z '&' (c 'or' a).z '&' ('not'( a '&' b '&' c)).z by MARGREL1:def 20 .=(a 'or' b).z '&' (b 'or' c).z '&' (c 'or' a).z '&' ('not'( a '&' b '&' c)).z by MARGREL1:def 20 .=(a 'or' b).z '&' (b 'or' c).z '&' (c 'or' a).z '&' (('not' a 'or' 'not' b 'or' 'not' c)).z by BVFUNC_5:37 .=(a.z 'or' b.z) '&' (b 'or' c).z '&' (c 'or' a).z '&' (('not' a 'or' 'not' b 'or' 'not' c)).z by BVFUNC_1:def 4 .=(a.z 'or' b.z) '&' (b.z 'or' c.z) '&' (c 'or' a).z '&' (( 'not' a 'or' 'not' b 'or' 'not' c)).z by BVFUNC_1:def 4 .=(a.z 'or' b.z) '&' (b.z 'or' c.z) '&' (c.z 'or' a.z) '&' (('not' a 'or' 'not' b 'or' 'not' c)).z by BVFUNC_1:def 4 .=(a.z 'or' b.z) '&' (b.z 'or' c.z) '&' (c.z 'or' a.z) '&' (('not' a 'or' 'not' b).z 'or' ('not' c).z) by BVFUNC_1:def 4 .=(a.z 'or' b.z) '&' (b.z 'or' c.z) '&' (c.z 'or' a.z) '&' (('not' a).z 'or' ('not' b).z 'or' ('not' c).z) by BVFUNC_1:def 4 .=(a.z 'or' b.z) '&' (b.z 'or' c.z) '&' (c.z 'or' a.z) '&' ('not' a.z 'or' ('not' b).z 'or' ('not' c).z) by MARGREL1:def 19 .=(a.z 'or' b.z) '&' (b.z 'or' c.z) '&' (c.z 'or' a.z) '&' ('not' a.z 'or' 'not' b.z 'or' ('not' c).z) by MARGREL1:def 19 .=(a.z 'or' b.z) '&' (b.z 'or' c.z) '&' (c.z 'or' a.z) '&' ('not' a.z 'or' 'not' b.z 'or' 'not' c.z) by MARGREL1:def 19; assume A3: ((a 'or' b) '&' (b 'or' c) '&' (c 'or' a) '&' 'not' (a '&' b '&' c )).z=TRUE; now A4: (a.z '&' b.z '&' 'not' c.z)=TRUE or (a.z '&' b.z '&' 'not' c.z)=FALSE by XBOOLEAN:def 3; assume A5: (('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c)).z<>TRUE; A6: (a.z '&' 'not' b.z '&' c.z)=TRUE or (a.z '&' 'not' b.z '&' c.z)=FALSE by XBOOLEAN:def 3; A7: (('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c)).z =(('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c)).z 'or' (a '&' b '&' 'not' c).z by BVFUNC_1:def 4 .=('not' a '&' b '&' c).z 'or' (a '&' 'not' b '&' c).z 'or' (a '&' b '&' 'not' c).z by BVFUNC_1:def 4 .=('not' a '&' b '&' c).z 'or' (a '&' 'not' b '&' c).z 'or' ((a '&' b).z '&' ('not' c).z) by MARGREL1:def 20 .=('not' a '&' b '&' c).z 'or' (a '&' 'not' b '&' c).z 'or' (a.z '&' b.z '&' ('not' c).z) by MARGREL1:def 20 .=('not' a '&' b '&' c).z 'or' ((a '&' 'not' b).z '&' c.z) 'or' (( a).z '&' b.z '&' ('not' c).z) by MARGREL1:def 20 .=('not' a '&' b '&' c).z 'or' (a.z '&' ('not' b).z '&' c.z) 'or' (a.z '&' b.z '&' ('not' c).z) by MARGREL1:def 20 .=(('not' a '&' b).z '&' c.z) 'or' (a.z '&' ('not' b).z '&' c. z) 'or' (a.z '&' b.z '&' ('not' c).z) by MARGREL1:def 20 .=(('not' a).z '&' b.z '&' c.z) 'or' (a.z '&' ('not' b).z '&' c.z) 'or' (a.z '&' b.z '&' ('not' c).z) by MARGREL1:def 20 .=('not' a.z '&' b.z '&' c.z) 'or' (a.z '&' ('not' b).z '&' c.z) 'or' (a.z '&' b.z '&' ('not' c).z) by MARGREL1:def 19 .=('not' a.z '&' b.z '&' c.z) 'or' (a.z '&' 'not' b.z '&' c.z) 'or' (a.z '&' b.z '&' ('not' c).z) by MARGREL1:def 19 .=('not' a.z '&' b.z '&' c.z) 'or' (a.z '&' 'not' b.z '&' c.z) 'or' (a.z '&' b.z '&' 'not' c.z) by MARGREL1:def 19; A8: ('not' a.z '&' b.z '&' c.z) 'or' (a.z '&' 'not' b.z '&' (c ).z)=TRUE or ('not' a.z '&' b.z '&' c.z) 'or' (a.z '&' 'not' b.z '&' c.z)=FALSE by XBOOLEAN:def 3; now per cases by A5,A7,A8,A6,MARGREL1:12; case A9: 'not' a.z '&' b.z=FALSE; now per cases by A9,MARGREL1:12; case A10: 'not' a.z=FALSE; now per cases by A5,A7,A6,A10,MARGREL1:12; case 'not' b.z=FALSE; hence thesis by A3,A2,A7,A10; end; case c.z=FALSE; hence thesis by A3,A2,A7,A10; end; end; hence thesis; end; case A11: b.z=FALSE; now per cases by A5,A7,A6,A11,MARGREL1:12; case a.z=FALSE; hence thesis by A3,A2,A11; end; case c.z=FALSE; hence thesis by A3,A2,A11; end; end; hence thesis; end; end; hence thesis; end; case A12: c.z=FALSE; now per cases by A5,A7,A4,A12,MARGREL1:12; case a.z=FALSE; hence thesis by A3,A2,A12; end; case b.z=FALSE; hence thesis by A3,A2,A12; end; end; hence thesis; end; end; hence thesis; end; hence thesis; end; then A13: (a 'or' b) '&' (b 'or' c) '&' (c 'or' a) '&' 'not'( a '&' b '&' c) '<' ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c); ('not' c '&' a '&' b) 'or' (c '&' 'not' a '&' b) 'or' (c '&' a '&' 'not' b) '<' (c 'or' a) '&' 'not'( c '&' a '&' b) by Lm4; then ('not' c '&' a '&' b) 'or' (c '&' 'not' a '&' b) 'or' (c '&' a '&' 'not' b) '<' (c 'or' a) '&' 'not'( a '&' b '&' c) by BVFUNC_1:4; then (a '&' b '&' 'not' c) 'or' (c '&' 'not' a '&' b) 'or' (c '&' a '&' 'not' b) '<' (c 'or' a) '&' 'not'( a '&' b '&' c) by BVFUNC_1:4; then (a '&' b '&' 'not' c) 'or' ('not' a '&' b '&' c) 'or' (c '&' a '&' 'not' b) '<' (c 'or' a) '&' 'not'( a '&' b '&' c) by BVFUNC_1:4; then (a '&' b '&' 'not' c) 'or' ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) '<' (c 'or' a) '&' 'not'( a '&' b '&' c) by BVFUNC_1:4; then ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c) '<' (c 'or' a) '&' 'not'( a '&' b '&' c) by BVFUNC_1:8; then A14: ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c) 'imp' (c 'or' a) '&' 'not'( a '&' b '&' c) = I_el(Y) by BVFUNC_1:16; ('not' b '&' c '&' a) 'or' (b '&' 'not' c '&' a) 'or' (b '&' c '&' 'not' a) '<' (b 'or' c) '&' 'not'( b '&' c '&' a) by Lm4; then ('not' b '&' c '&' a) 'or' (b '&' 'not' c '&' a) 'or' (b '&' c '&' 'not' a) '<' (b 'or' c) '&' 'not'( a '&' b '&' c) by BVFUNC_1:4; then ('not' b '&' c '&' a) 'or' (b '&' 'not' c '&' a) 'or' ('not' a '&' b '&' c) '<' (b 'or' c) '&' 'not'( a '&' b '&' c) by BVFUNC_1:4; then (a '&' 'not' b '&' c) 'or' (b '&' 'not' c '&' a) 'or' ('not' a '&' b '&' c) '<' (b 'or' c) '&' 'not'( a '&' b '&' c) by BVFUNC_1:4; then (a '&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c) 'or' ('not' a '&' b '&' c) '<' (b 'or' c) '&' 'not'( a '&' b '&' c) by BVFUNC_1:4; then ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c) '<' (b 'or' c) '&' 'not'( a '&' b '&' c) by BVFUNC_1:8; then A15: ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c) 'imp' (b 'or' c) '&' 'not'( a '&' b '&' c) = I_el(Y) by BVFUNC_1:16; ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c) '<' (a 'or' b) '&' 'not'( a '&' b '&' c) by Lm4; then ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c) 'imp' (a 'or' b) '&' 'not'( a '&' b '&' c) = I_el(Y) by BVFUNC_1:16; then ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c) 'imp' (a 'or' b) '&' 'not'( a '&' b '&' c) '&' ((b 'or' c) '&' 'not'( a '&' b '&' c)) = I_el(Y) by A15,th18; then ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c) 'imp' (a 'or' b) '&' 'not'( a '&' b '&' c) '&' (b 'or' c) '&' 'not'( a '&' b '&' c) = I_el(Y) by BVFUNC_1:4; then ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c) 'imp' (a 'or' b) '&' (b 'or' c) '&' 'not'( a '&' b '&' c) '&' 'not'( a '&' b '&' c) = I_el(Y) by BVFUNC_1:4; then ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c) 'imp' (a 'or' b) '&' (b 'or' c) '&' ('not'( a '&' b '&' c) '&' 'not'( a '&' b '&' c)) = I_el(Y) by BVFUNC_1:4; then ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c) 'imp' (a 'or' b) '&' (b 'or' c) '&' 'not'( a '&' b '&' c) '&' ((c 'or' a) '&' 'not'( a '&' b '&' c)) = I_el(Y) by A14,th18; then ('not' a '&' b '&' c) 'or' (a '&' 'not' b '&' c) 'or' (a '&' b '&' 'not' c) '<' (a 'or' b) '&' (b 'or' c) '&' (c 'or' a) '&' 'not'( a '&' b '&' c) by A1,BVFUNC_1:16; hence thesis by A13,BVFUNC_1:15; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '<' (a 'imp' (b '&' c)) proof let a,b,c be Function of Y,BOOLEAN; let z be Element of Y; A1: ((a 'imp' b) '&' (b 'imp' c)).z =(a 'imp' b).z '&' (b 'imp' c).z by MARGREL1:def 20 .=('not' a 'or' b).z '&' (b 'imp' c).z by BVFUNC_4:8 .=('not' a 'or' b).z '&' ('not' b 'or' c).z by BVFUNC_4:8 .=(('not' a).z 'or' b.z) '&' ('not' b 'or' c).z by BVFUNC_1:def 4 .=(('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) by BVFUNC_1:def 4; assume A2: ((a 'imp' b) '&' (b 'imp' c)).z=TRUE; now A3: (a 'imp' (b '&' c)).z =('not' a 'or' (b '&' c)).z by BVFUNC_4:8 .=('not' a).z 'or' (b '&' c).z by BVFUNC_1:def 4 .=('not' a).z 'or' (b.z '&' c.z) by MARGREL1:def 20 .='not' a.z 'or' (b.z '&' c.z) by MARGREL1:def 19; assume A4: (a 'imp' (b '&' c)).z<>TRUE; 'not' a.z=TRUE or 'not' a.z=FALSE by XBOOLEAN:def 3; then A5: ('not' a).z=FALSE by A4,A3,MARGREL1:def 19; A6: (b.z '&' c.z)=TRUE or (b.z '&' c.z)=FALSE by XBOOLEAN:def 3; now per cases by A4,A3,A6,MARGREL1:12; case b.z=FALSE; hence thesis by A2,A1,A5; end; case c.z=FALSE; then (('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) =b.z '&' 'not' b.z by A5,MARGREL1:def 19 .=FALSE by XBOOLEAN:138; hence thesis by A2,A1; end; end; hence thesis; end; hence thesis; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '<' ((a 'or' b) 'imp' c) proof let a,b,c be Function of Y,BOOLEAN; let z be Element of Y; A1: ((a 'imp' b) '&' (b 'imp' c)).z =(a 'imp' b).z '&' (b 'imp' c).z by MARGREL1:def 20 .=('not' a 'or' b).z '&' (b 'imp' c).z by BVFUNC_4:8 .=('not' a 'or' b).z '&' ('not' b 'or' c).z by BVFUNC_4:8 .=(('not' a).z 'or' b.z) '&' ('not' b 'or' c).z by BVFUNC_1:def 4 .=(('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) by BVFUNC_1:def 4; assume A2: ((a 'imp' b) '&' (b 'imp' c)).z=TRUE; now A3: c.z=TRUE or c.z=FALSE by XBOOLEAN:def 3; assume A4: ((a 'or' b) 'imp' c).z<>TRUE; A5: ((a 'or' b) 'imp' c).z =('not'( a 'or' b) 'or' c).z by BVFUNC_4:8 .=(('not' a '&' 'not' b) 'or' c).z by BVFUNC_1:13 .=('not' a '&' 'not' b).z 'or' c.z by BVFUNC_1:def 4 .=(('not' a).z '&' ('not' b).z) 'or' c.z by MARGREL1:def 20; A6: (('not' a).z '&' ('not' b).z)=TRUE or (('not' a).z '&' ('not' b).z)= FALSE by XBOOLEAN:def 3; now per cases by A4,A5,A6,MARGREL1:12; case ('not' a).z=FALSE; then (('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) =b.z '&' 'not' b.z by A4,A5,A3,MARGREL1:def 19 .=FALSE by XBOOLEAN:138; hence thesis by A2,A1; end; case ('not' b).z=FALSE; then (('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) =(('not' a) .z 'or' b.z) '&' FALSE by A4,A5,XBOOLEAN:def 3 .=FALSE; hence thesis by A2,A1; end; end; hence thesis; end; hence thesis; end; theorem Th19: for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '<' (a 'imp' (b 'or' c)) proof let a,b,c be Function of Y,BOOLEAN; let z be Element of Y; A1: ((a 'imp' b) '&' (b 'imp' c)).z =(a 'imp' b).z '&' (b 'imp' c).z by MARGREL1:def 20 .=('not' a 'or' b).z '&' (b 'imp' c).z by BVFUNC_4:8 .=('not' a 'or' b).z '&' ('not' b 'or' c).z by BVFUNC_4:8 .=(('not' a).z 'or' b.z) '&' ('not' b 'or' c).z by BVFUNC_1:def 4 .=(('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) by BVFUNC_1:def 4; assume A2: ((a 'imp' b) '&' (b 'imp' c)).z=TRUE; now assume (a 'imp' (b 'or' c)).z<>TRUE; A3: ('not' a).z=TRUE or ('not' a).z=FALSE by XBOOLEAN:def 3; A4: (b.z 'or' c.z)=TRUE or (b.z 'or' c.z)=FALSE by XBOOLEAN:def 3; A5: c.z=TRUE or c.z=FALSE by XBOOLEAN:def 3; (a 'imp' (b 'or' c)).z =('not' a 'or' (b 'or' c)).z by BVFUNC_4:8 .=('not' a).z 'or' (b 'or' c).z by BVFUNC_1:def 4 .=('not' a).z 'or' (b.z 'or' c.z) by BVFUNC_1:def 4; hence thesis by A2,A1,A3,A4,A5; end; hence thesis; end; theorem Th20: for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '<' (a 'imp' (b 'or' 'not' c)) proof let a,b,c be Function of Y,BOOLEAN; let z be Element of Y; A1: ((a 'imp' b) '&' (b 'imp' c)).z =(a 'imp' b).z '&' (b 'imp' c).z by MARGREL1:def 20 .=('not' a 'or' b).z '&' (b 'imp' c).z by BVFUNC_4:8 .=('not' a 'or' b).z '&' ('not' b 'or' c).z by BVFUNC_4:8 .=(('not' a).z 'or' b.z) '&' ('not' b 'or' c).z by BVFUNC_1:def 4 .=(('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) by BVFUNC_1:def 4; assume A2: ((a 'imp' b) '&' (b 'imp' c)).z=TRUE; now assume (a 'imp' (b 'or' 'not' c)).z<>TRUE; A3: ('not' a).z=TRUE or ('not' a).z=FALSE by XBOOLEAN:def 3; A4: (b.z 'or' ('not' c).z)=TRUE or (b.z 'or' ('not' c).z)= FALSE by XBOOLEAN:def 3; A5: ('not' c).z=TRUE or ('not' c).z=FALSE by XBOOLEAN:def 3; (a 'imp' (b 'or' 'not' c)).z =('not' a 'or' (b 'or' 'not' c)).z by BVFUNC_4:8 .=('not' a).z 'or' (b 'or' 'not' c).z by BVFUNC_1:def 4 .=('not' a).z 'or' (b.z 'or' ('not' c).z) by BVFUNC_1:def 4; hence thesis by A2,A1,A3,A4,A5; end; hence thesis; end; theorem Th21: for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '<' (b 'imp' (c 'or' a)) proof let a,b,c be Function of Y,BOOLEAN; let z be Element of Y; A1: ((a 'imp' b) '&' (b 'imp' c)).z =(a 'imp' b).z '&' (b 'imp' c).z by MARGREL1:def 20 .=('not' a 'or' b).z '&' (b 'imp' c).z by BVFUNC_4:8 .=('not' a 'or' b).z '&' ('not' b 'or' c).z by BVFUNC_4:8 .=(('not' a).z 'or' b.z) '&' ('not' b 'or' c).z by BVFUNC_1:def 4 .=(('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) by BVFUNC_1:def 4; assume A2: ((a 'imp' b) '&' (b 'imp' c)).z=TRUE; now assume (b 'imp' (c 'or' a)).z<>TRUE; A3: ('not' b).z=TRUE or ('not' b).z=FALSE by XBOOLEAN:def 3; A4: (c.z 'or' a.z)=TRUE or (c.z 'or' a.z)=FALSE by XBOOLEAN:def 3; A5: a.z=TRUE or a.z=FALSE by XBOOLEAN:def 3; (b 'imp' (c 'or' a)).z =('not' b 'or' (c 'or' a)).z by BVFUNC_4:8 .=('not' b).z 'or' (c 'or' a).z by BVFUNC_1:def 4 .=('not' b).z 'or' (c.z 'or' a.z) by BVFUNC_1:def 4; hence thesis by A2,A1,A3,A4,A5; end; hence thesis; end; theorem Th22: for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '<' (b 'imp' (c 'or' 'not' a)) proof let a,b,c be Function of Y,BOOLEAN; let z be Element of Y; A1: ((a 'imp' b) '&' (b 'imp' c)).z =(a 'imp' b).z '&' (b 'imp' c).z by MARGREL1:def 20 .=('not' a 'or' b).z '&' (b 'imp' c).z by BVFUNC_4:8 .=('not' a 'or' b).z '&' ('not' b 'or' c).z by BVFUNC_4:8 .=(('not' a).z 'or' b.z) '&' ('not' b 'or' c).z by BVFUNC_1:def 4 .=(('not' a).z 'or' b.z) '&' (('not' b).z 'or' c.z) by BVFUNC_1:def 4; assume A2: ((a 'imp' b) '&' (b 'imp' c)).z=TRUE; now assume (b 'imp' (c 'or' 'not' a)).z<>TRUE; A3: ('not' b).z=TRUE or ('not' b).z=FALSE by XBOOLEAN:def 3; A4: (c.z 'or' ('not' a).z)=TRUE or (c.z 'or' ('not' a).z)= FALSE by XBOOLEAN:def 3; A5: ('not' a).z=TRUE or ('not' a).z=FALSE by XBOOLEAN:def 3; (b 'imp' (c 'or' 'not' a)).z =('not' b 'or' (c 'or' 'not' a)).z by BVFUNC_4:8 .=('not' b).z 'or' (c 'or' 'not' a).z by BVFUNC_1:def 4 .=('not' b).z 'or' (c.z 'or' ('not' a).z) by BVFUNC_1:def 4; hence thesis by A2,A1,A3,A4,A5; end; hence thesis; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '<' (a 'imp' b) '&' (b 'imp' (c 'or' a)) proof let a,b,c be Function of Y,BOOLEAN; (a 'imp' b) '&' (b 'imp' c) '<' (b 'imp' (c 'or' a)) by Th21; then A1: (a 'imp' b) '&' (b 'imp' c) 'imp' (b 'imp' (c 'or' a))=I_el(Y) by BVFUNC_1:16; (a 'imp' b) '&' (b 'imp' c) 'imp' (a 'imp' b)=I_el(Y) by Th38; then (a 'imp' b) '&' (b 'imp' c) 'imp' (a 'imp' b) '&' (b 'imp' (c 'or' a))= I_el(Y) by A1,th18; hence thesis by BVFUNC_1:16; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '<' (a 'imp' (b 'or' 'not' c)) '&' (b 'imp' c) proof let a,b,c be Function of Y,BOOLEAN; (a 'imp' b) '&' (b 'imp' c) '<' (a 'imp' (b 'or' 'not' c)) by Th20; then A1: (a 'imp' b) '&' (b 'imp' c) 'imp' (a 'imp' (b 'or' 'not' c))=I_el(Y) by BVFUNC_1:16; (a 'imp' b) '&' (b 'imp' c) 'imp' (b 'imp' c)=I_el(Y) by Th38; then (a 'imp' b) '&' (b 'imp' c) 'imp' (a 'imp' (b 'or' 'not' c)) '&' (b 'imp' c)=I_el(Y) by A1,th18; hence thesis by BVFUNC_1:16; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '<' (a 'imp' (b 'or' c)) '&' (b 'imp' (c 'or' a)) proof let a,b,c be Function of Y,BOOLEAN; (a 'imp' b) '&' (b 'imp' c) '<' (b 'imp' (c 'or' a)) by Th21; then A1: (a 'imp' b) '&' (b 'imp' c) 'imp' (b 'imp' (c 'or' a))=I_el(Y) by BVFUNC_1:16; (a 'imp' b) '&' (b 'imp' c) '<' (a 'imp' (b 'or' c)) by Th19; then (a 'imp' b) '&' (b 'imp' c) 'imp' (a 'imp' (b 'or' c))=I_el(Y) by BVFUNC_1:16; then (a 'imp' b) '&' (b 'imp' c) 'imp' (a 'imp' (b 'or' c)) '&' (b 'imp' (c 'or' a))=I_el(Y) by A1,th18; hence thesis by BVFUNC_1:16; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '<' (a 'imp' (b 'or' 'not' c)) '&' (b 'imp' (c 'or' a)) proof let a,b,c be Function of Y,BOOLEAN; (a 'imp' b) '&' (b 'imp' c) '<' (b 'imp' (c 'or' a)) by Th21; then A1: (a 'imp' b) '&' (b 'imp' c) 'imp' (b 'imp' (c 'or' a))=I_el(Y) by BVFUNC_1:16; (a 'imp' b) '&' (b 'imp' c) '<' (a 'imp' (b 'or' 'not' c)) by Th20; then (a 'imp' b) '&' (b 'imp' c) 'imp' (a 'imp' (b 'or' 'not' c))=I_el(Y) by BVFUNC_1:16; then (a 'imp' b) '&' (b 'imp' c) 'imp' (a 'imp' (b 'or' 'not' c)) '&' (b 'imp' (c 'or' a))=I_el(Y) by A1,th18; hence thesis by BVFUNC_1:16; end; theorem for a,b,c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '<' (a 'imp' (b 'or' 'not' c)) '&' (b 'imp' (c 'or' 'not' a)) proof let a,b,c be Function of Y,BOOLEAN; (a 'imp' b) '&' (b 'imp' c) '<' (b 'imp' (c 'or' 'not' a)) by Th22; then A1: (a 'imp' b) '&' (b 'imp' c) 'imp' (b 'imp' (c 'or' 'not' a))=I_el(Y) by BVFUNC_1:16; (a 'imp' b) '&' (b 'imp' c) '<' (a 'imp' (b 'or' 'not' c)) by Th20; then (a 'imp' b) '&' (b 'imp' c) 'imp' (a 'imp' (b 'or' 'not' c))=I_el(Y) by BVFUNC_1:16; then (a 'imp' b) '&' (b 'imp' c) 'imp' (a 'imp' (b 'or' 'not' c)) '&' (b 'imp' (c 'or' 'not' a))=I_el(Y) by A1,th18; hence thesis by BVFUNC_1:16; end;