### Propositional Calculus for Boolean Valued Functions. Part IV

by
Shunichi Kobayashi

Copyright (c) 1999 Association of Mizar Users

MML identifier: BVFUNC_8
[ MML identifier index ]

```environ

vocabulary FUNCT_2, MARGREL1, BVFUNC_1, ZF_LANG, FUNCT_1, RELAT_1, BINARITH,
PARTIT1;
notation TARSKI, XBOOLE_0, SUBSET_1, MARGREL1, VALUAT_1, RELAT_1, FUNCT_1,
FRAENKEL, BINARITH, BVFUNC_1;
constructors BINARITH, BVFUNC_1;
clusters MARGREL1, VALUAT_1, BINARITH, FRAENKEL;
requirements SUBSET, BOOLE;
definitions BVFUNC_1;
theorems FUNCT_1, FUNCT_2, MARGREL1, BINARITH, BVFUNC_1, BVFUNC_4, VALUAT_1;

begin

::Chap. 1  Propositional Calculus

reserve Y for non empty set;

theorem for a,b,c,d being Element of Funcs(Y,BOOLEAN) holds
a 'imp' (b '&' c '&' d) = (a 'imp' b) '&' (a 'imp' c) '&' (a 'imp' d)
proof
let a,b,c,d be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj(a 'imp' (b '&' c '&' d),x) =
Pj((a 'imp' b) '&' (a 'imp' c) '&' (a 'imp' d),x)
proof
let x be Element of Y;
Pj((a 'imp' b) '&' (a 'imp' c) '&' (a 'imp' d),x)
=Pj((a 'imp' b) '&' (a 'imp' c),x) '&' Pj(a 'imp' d,x) by VALUAT_1:def 6
.=Pj(a 'imp' b,x) '&' Pj(a 'imp' c,x) '&' Pj(a 'imp' d,x) by VALUAT_1:def 6
.=('not' Pj(a,x) 'or' Pj(b,x)) '&' Pj(a 'imp' c,x) '&' Pj(a 'imp' d,x)
by BVFUNC_1:def 11
.=('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not'
Pj(a,x) 'or' Pj(c,x)) '&' Pj(a 'imp' d,x)
by BVFUNC_1:def 11
.=('not' Pj(a,x) 'or' Pj(b,x)) '&' ('not' Pj(a,x) 'or' Pj(c,x)) '&'
('not' Pj(a,x) 'or' Pj(d,x))
by BVFUNC_1:def 11
.=('not' Pj(a,x) 'or' (Pj(b,x) '&' Pj(c,x))) '&' ('not' Pj(a,x) 'or' Pj(d,x))
by BINARITH:23
.='not' Pj(a,x) 'or' (Pj(b,x) '&' Pj(c,x) '&' Pj(d,x))
by BINARITH:23
.='not' Pj(a,x) 'or' (Pj(b '&' c,x) '&' Pj(d,x)) by VALUAT_1:def 6
.='not' Pj(a,x) 'or' (Pj(b '&' c '&' d,x)) by VALUAT_1:def 6
.=Pj(a 'imp' (b '&' c '&' d),x) by BVFUNC_1:def 11;
hence thesis;
end;
consider k3 being Function such that
A2: (a 'imp' (b '&' c '&' d))=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: (a 'imp' b) '&' (a 'imp' c) '&' (a 'imp' d)=
k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence thesis by A2,A3,FUNCT_1:9;
end;

theorem for a,b,c,d being Element of Funcs(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 Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj(a 'imp' (b 'or' c 'or' d),x) =
Pj((a 'imp' b) 'or' (a 'imp' c) 'or' (a 'imp' d),x)
proof
let x be Element of Y;
Pj((a 'imp' b) 'or' (a 'imp' c) 'or' (a 'imp' d),x)
=Pj((a 'imp' b) 'or' (a 'imp' c),x) 'or' Pj(a 'imp' d,x)
by BVFUNC_1:def 7
.=Pj(a 'imp' b,x) 'or' Pj(a 'imp' c,x) 'or' Pj(a 'imp' d,x)
by BVFUNC_1:def 7
.=('not' Pj(a,x) 'or' Pj(b,x)) 'or' Pj(a 'imp' c,x) 'or' Pj(a 'imp' d,x)
by BVFUNC_1:def 11
.=('not' Pj(a,x) 'or' Pj(b,x)) 'or' ('not' Pj(a,x) 'or' Pj(c,x)) 'or'
Pj(a 'imp' d,x)
by BVFUNC_1:def 11
.=('not' Pj(a,x) 'or' Pj(b,x)) 'or' ('not' Pj(a,x) 'or' Pj(c,x))
'or' ('not' Pj(a,x) 'or' Pj(d,x))
by BVFUNC_1:def 11
.=(('not' Pj(a,x) 'or' ('not' Pj(a,x) 'or' Pj(b,x))) 'or' Pj(c,x))
'or' ('not' Pj(a,x) 'or' Pj(d,x)) by BINARITH:20
.=((('not' Pj(a,x) 'or' 'not' Pj(a,x)) 'or' Pj(b,x)) 'or' Pj(c,x))
'or' ('not' Pj(a,x) 'or' Pj(d,x))
by BINARITH:20
.=(('not' Pj(a,x) 'or' Pj(b,x)) 'or' Pj(c,x))
'or' ('not' Pj(a,x) 'or' Pj(d,x))
by BINARITH:21
.=('not' Pj(a,x) 'or' (Pj(b,x) 'or' Pj(c,x)))
'or' ('not' Pj(a,x) 'or' Pj(d,x))
by BINARITH:20
.=('not' Pj(a,x) 'or' Pj(b 'or' c,x)) 'or' ('not' Pj(a,x) 'or' Pj(d,x))
by BVFUNC_1:def 7
.=('not' Pj(a,x) 'or' ('not'
Pj(a,x) 'or' Pj(b 'or' c,x))) 'or' Pj(d,x) by BINARITH:20
.=(('not' Pj(a,x) 'or' 'not' Pj(a,x)) 'or' Pj(b 'or' c,x)) 'or' Pj(d,x)
by BINARITH:20
.=('not' Pj(a,x) 'or' Pj(b 'or' c,x)) 'or' Pj(d,x)
by BINARITH:21
.='not' Pj(a,x) 'or' (Pj(b 'or' c,x) 'or' Pj(d,x))
by BINARITH:20
.='not' Pj(a,x) 'or' Pj(b 'or' c 'or' d,x)
by BVFUNC_1:def 7
.=Pj(a 'imp' (b 'or' c 'or' d),x) by BVFUNC_1:def 11;
hence thesis;
end;
consider k3 being Function such that
A2: (a 'imp' (b 'or' c 'or' d))=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: (a 'imp' b) 'or' (a 'imp' c) 'or' (a 'imp' d)=
k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence thesis by A2,A3,FUNCT_1:9;
end;

theorem for a,b,c,d being Element of Funcs(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 Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj((a '&' b '&' c) 'imp' d,x) =
Pj((a 'imp' d) 'or' (b 'imp' d) 'or' (c 'imp' d),x)
proof
let x be Element of Y;
Pj((a 'imp' d) 'or' (b 'imp' d) 'or' (c 'imp' d),x)
=Pj((a 'imp' d) 'or' (b 'imp' d),x) 'or' Pj(c 'imp' d,x)
by BVFUNC_1:def 7
.=Pj(a 'imp' d,x) 'or' Pj(b 'imp' d,x) 'or' Pj(c 'imp' d,x)
by BVFUNC_1:def 7
.=('not' Pj(a,x) 'or' Pj(d,x)) 'or' Pj(b 'imp' d,x) 'or' Pj(c 'imp' d,x)
by BVFUNC_1:def 11
.=('not' Pj(a,x) 'or' Pj(d,x)) 'or' ('not' Pj(b,x) 'or' Pj(d,x))
'or' Pj(c 'imp' d,x)
by BVFUNC_1:def 11
.=('not' Pj(a,x) 'or' Pj(d,x)) 'or' ('not' Pj(b,x) 'or' Pj(d,x))
'or' ('not' Pj(c,x) 'or' Pj(d,x))
by BVFUNC_1:def 11
.=(('not' Pj(a,x) 'or' (Pj(d,x) 'or' 'not' Pj(b,x))) 'or' Pj(d,x))
'or' ('not' Pj(c,x) 'or' Pj(d,x))
by BINARITH:20
.=(('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(d,x)) 'or' Pj(d,x)
'or' ('not' Pj(c,x) 'or' Pj(d,x))
by BINARITH:20
.=('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' (Pj(d,x) 'or' Pj(d,x))
'or' ('not' Pj(c,x) 'or' Pj(d,x))
by BINARITH:20
.=('not' Pj(a,x) 'or' 'not' Pj(b,x)) 'or' Pj(d,x)
'or' ('not' Pj(c,x) 'or' Pj(d,x))
by BINARITH:21
.=('not'( Pj(a,x) '&' Pj(b,x)) 'or' Pj(d,x)) 'or'
('not' Pj(c,x) 'or' Pj(d,x))
by BINARITH:9
.=('not'( Pj(a,x) '&' Pj(b,x)) 'or' (Pj(d,x) 'or' 'not'
Pj(c,x))) 'or' Pj(d,x)
by BINARITH:20
.=(('not'( Pj(a,x) '&' Pj(b,x)) 'or' 'not' Pj(c,x)) 'or'
Pj(d,x)) 'or' Pj(d,x) by BINARITH:20
.=('not'( Pj(a,x) '&' Pj(b,x)) 'or' 'not' Pj(c,x)) 'or'
(Pj(d,x) 'or' Pj(d,x)) by BINARITH:20
.=('not'( Pj(a,x) '&' Pj(b,x)) 'or' 'not' Pj(c,x)) 'or' Pj(d,x)
by BINARITH:21
.='not'( Pj(a,x) '&' Pj(b,x) '&' Pj(c,x)) 'or' Pj(d,x) by BINARITH:9
.='not'( Pj(a '&' b,x) '&' Pj(c,x)) 'or' Pj(d,x) by VALUAT_1:def 6
.='not' Pj(a '&' b '&' c,x) 'or' Pj(d,x) by VALUAT_1:def 6
.=Pj((a '&' b '&' c) 'imp' d,x) by BVFUNC_1:def 11;
hence thesis;
end;
consider k3 being Function such that
A2: ((a '&' b '&' c) 'imp' d)=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: (a 'imp' d) 'or' (b 'imp' d) 'or' (c 'imp' d)
=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence thesis by A2,A3,FUNCT_1:9;
end;

theorem for a,b,c,d being Element of Funcs(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 Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj((a 'or' b 'or' c) 'imp' d,x) =
Pj((a 'imp' d) '&' (b 'imp' d) '&' (c 'imp' d),x)
proof
let x be Element of Y;
Pj((a 'imp' d) '&' (b 'imp' d) '&' (c 'imp' d),x)
=Pj((a 'imp' d) '&' (b 'imp' d),x) '&' Pj(c 'imp' d,x)
by VALUAT_1:def 6
.=Pj(a 'imp' d,x) '&' Pj(b 'imp' d,x) '&' Pj(c 'imp' d,x)
by VALUAT_1:def 6
.=('not' Pj(a,x) 'or' Pj(d,x)) '&' Pj(b 'imp' d,x) '&' Pj(c 'imp' d,x)
by BVFUNC_1:def 11
.=('not' Pj(a,x) 'or' Pj(d,x)) '&' ('not'
Pj(b,x) 'or' Pj(d,x)) '&' Pj(c 'imp' d,x)
by BVFUNC_1:def 11
.=(Pj(d,x) 'or' 'not' Pj(a,x)) '&' ('not' Pj(b,x) 'or' Pj(d,x))
'&' ('not' Pj(c,x) 'or' Pj(d,x)) by BVFUNC_1:def 11
.=(Pj(d,x) 'or' ('not' Pj(a,x) '&' 'not' Pj(b,x)))
'&' ('not' Pj(c,x) 'or' Pj(d,x))
by BINARITH:23
.=('not'( Pj(a,x) 'or' Pj(b,x)) 'or' Pj(d,x))
'&' ('not' Pj(c,x) 'or' Pj(d,x))
by BINARITH:10
.=(Pj(d,x) 'or' 'not' Pj(a 'or' b,x)) '&' ('not' Pj(c,x) 'or' Pj(d,x)) by
BVFUNC_1:def 7
.=Pj(d,x) 'or' ('not' Pj(a 'or' b,x) '&' 'not' Pj(c,x))
by BINARITH:23
.=('not'( Pj(a 'or' b,x) 'or' Pj(c,x))) 'or' Pj(d,x)
by BINARITH:10
.='not' Pj(a 'or' b 'or' c,x) 'or' Pj(d,x)
by BVFUNC_1:def 7
.=Pj((a 'or' b 'or' c) 'imp' d,x) by BVFUNC_1:def 11;
hence thesis;
end;
consider k3 being Function such that
A2: ((a 'or' b 'or' c) 'imp' d)=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: (a 'imp' d) '&' (b 'imp' d) '&' (c 'imp' d)
=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence thesis by A2,A3,FUNCT_1:9;
end;

theorem for a,b,c being Element of Funcs(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 Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj((a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a),x) =
Pj((a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a) '&'
(b 'imp' a) '&' (a 'imp' c),x)
proof
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_1:15
.=(('not' a '&' ('not' b 'or' c) 'or'
(b '&' 'not' b 'or' b '&' c))) '&' ('not' c 'or' a)
by BVFUNC_1:15
.=(('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_1:12
.=(('not' a 'or' (b '&' c)) '&' (('not' b 'or' c)
'or' (b '&' c))) '&' ('not' c 'or' a)
by BVFUNC_1:14
.=((('not' a 'or' b) '&' ('not' a 'or' c)) '&' (('not'
b 'or' c) 'or' (b '&' c)))
'&' ('not' c 'or' a)
by BVFUNC_1:14
.=((('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_1:14
.=((('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_1:11
.=((('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_1:13
.=((('not' a 'or' b) '&' ('not' a 'or' c)) '&'
(('not' b 'or' c) 'or' c))
'&' ('not' c 'or' a)
by BVFUNC_1:9
.=((('not' a 'or' b) '&' ('not' a 'or' c)) '&'
('not' b 'or' (c 'or' c)))
'&' ('not' c 'or' a)
by BVFUNC_1:11
.=((('not' a 'or' b) '&' ('not' a 'or' c)) '&' ('not' b 'or' c))
'&' ('not' c 'or' a)
by BVFUNC_1:10
.=(('not' a 'or' b) '&' ('not' a 'or' c)) '&' (('not' b 'or' c)
'&' ('not' c 'or' a))
by BVFUNC_1:7
.=(('not' a 'or' b) '&' ('not' a 'or' c)) '&'
(('not' b '&' ('not' c 'or' a)) 'or' (c '&' ('not' c 'or' a)))
by BVFUNC_1:15
.=(('not' a 'or' b) '&' ('not' a 'or' c)) '&'
(('not' b '&' ('not' c 'or' a)) 'or' ((c '&' 'not' c 'or' c '&' a)))
by BVFUNC_1:15
.=(('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_1:12
.=(('not' a 'or' b) '&' ('not' a 'or' c)) '&'
(('not' b 'or' (c '&' a)) '&' (('not' c 'or' a) 'or' (c '&' a)))
by BVFUNC_1:14
.=(('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_1:14
.= (('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_1:14
.= (('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_1:11
.= (('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_1:13
.= (('not' a 'or' c) '&' ('not' a 'or' b)) '&'
((('not' b 'or' a) '&' ('not' b 'or' c)) '&'
(('not' c 'or' a) 'or' a))
by BVFUNC_1:9
.= (('not' a 'or' c) '&' ('not' a 'or' b)) '&'
((('not' b 'or' a) '&' ('not' b 'or' c)) '&'
('not' c 'or' (a 'or' a)))
by BVFUNC_1:11
.= (('not' a 'or' c) '&' ('not' a 'or' b)) '&'
((('not' b 'or' a) '&' ('not' b 'or' c)) '&' ('not' c '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)
by BVFUNC_1:7
.= (((('not' a 'or' b) '&' ('not' a 'or' c)) '&' ('not' b 'or' a)) '&' ('not'
b 'or' c))
'&' ('not' c 'or' a)
by BVFUNC_1:7
.= (((('not' b 'or' a) '&' ('not' a 'or' c)) '&' ('not' a 'or' b)) '&' ('not'
b 'or' c))
'&' ('not' c 'or' a)
by BVFUNC_1:7
.= ((('not' b 'or' a) '&' ('not' a 'or' c)) '&' (('not' a 'or' b) '&' ('not'
b 'or' c)))
'&' ('not' c 'or' a)
by BVFUNC_1:7
.= (('not' b 'or' a) '&' ('not' a 'or' c)) '&'
((('not' a 'or' b) '&' ('not' b 'or' c)) '&' ('not' c 'or' a))
by BVFUNC_1:7
.= ('not' b 'or' a) '&'
(('not' a 'or' b) '&' ('not' b 'or' c) '&' ('not' c 'or' a)) '&'
('not' a 'or' c)
by BVFUNC_1:7
.= (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;
consider k3 being Function such that
A2: ((a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a))=
k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: (a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a) '&' (b 'imp' a) '&'
(a 'imp' c)=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence ((a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a))=
(a 'imp' b) '&' (b 'imp' c) '&' (c 'imp' a) '&'
(b 'imp' a) '&' (a 'imp' c) by A2,A3,FUNCT_1:9;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
a = (a '&' b) 'or' (a '&' 'not' b)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj(a,x) = Pj((a '&' b) 'or' (a '&' 'not' b),x)
proof
let x be Element of Y;
Pj((a '&' b) 'or' (a '&' 'not' b),x)
=Pj(a '&' (b 'or' 'not' b),x) by BVFUNC_1:15
.=Pj(a '&' I_el(Y),x) by BVFUNC_4:6
.=Pj(a,x) by BVFUNC_1:9;
hence thesis;
end;
consider k3 being Function such that
A2: (a)=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: (a '&' b) 'or' (a '&' 'not' b)
=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence thesis by A2,A3,FUNCT_1:9;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
a = (a 'or' b) '&' (a 'or' 'not' b)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj(a,x) = Pj((a 'or' b) '&' (a 'or' 'not' b),x)
proof
let x be Element of Y;
Pj((a 'or' b) '&' (a 'or' 'not' b),x)
=Pj(a 'or' (b '&' 'not' b),x) by BVFUNC_1:14
.=Pj(a 'or' O_el(Y),x) by BVFUNC_4:5
.=Pj(a,x) by BVFUNC_1:12;
hence thesis;
end;
consider k3 being Function such that
A2: (a)=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: (a 'or' b) '&' (a 'or' 'not' b)
=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence thesis by A2,A3,FUNCT_1:9;
end;

theorem for a,b,c being Element of Funcs(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 Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj(a,x) = Pj((a '&' b '&' c) 'or' (a '&' b '&' 'not' c) 'or'
(a '&' 'not' b '&' c) 'or' (a '&' 'not' b '&' 'not' c),x)
proof
let x be Element of Y;
Pj((a '&' b '&' c) 'or' (a '&' b '&' 'not' c) 'or'
(a '&' 'not' b '&' c) 'or' (a '&' 'not' b '&' 'not' c),x)
=Pj(((a '&' b) '&' (c 'or' 'not' c)) 'or'
(a '&' 'not' b '&' c) 'or' (a '&' 'not' b '&' 'not' c),x)
by BVFUNC_1:15
.=Pj(((a '&' b) '&' I_el(Y)) 'or'
(a '&' 'not' b '&' c) 'or' (a '&' 'not' b '&' 'not' c),x)
by BVFUNC_4:6
.=Pj((a '&' b) 'or'
(a '&' 'not' b '&' c) 'or' (a '&' 'not' b '&' 'not' c),x)
by BVFUNC_1:9
.=Pj((a '&' b) 'or'
((a '&' 'not' b '&' c) 'or' (a '&' 'not' b '&' 'not' c)),x)
by BVFUNC_1:11
.=Pj((a '&' b) 'or' ((a '&' 'not' b) '&' (c 'or' 'not' c)),x)
by BVFUNC_1:15
.=Pj((a '&' b) 'or' ((a '&' 'not' b) '&' I_el(Y)),x)
by BVFUNC_4:6
.=Pj((a '&' b) 'or' (a '&' 'not' b),x)
by BVFUNC_1:9
.=Pj(a '&' (b 'or' 'not' b),x) by BVFUNC_1:15
.=Pj(a '&' I_el(Y),x) by BVFUNC_4:6
.=Pj(a,x) by BVFUNC_1:9;
hence thesis;
end;
consider k3 being Function such that
A2: (a)=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: (a '&' b '&' c) 'or' (a '&' b '&' 'not' c) 'or' (a '&' 'not' b '&' c)
'or' (a '&' 'not' b '&' 'not' c)=
k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence (a)=(a '&' b '&' c) 'or' (a '&' b '&' 'not' c)
'or' (a '&' 'not' b '&' c) 'or' (a '&' 'not' b '&' 'not' c) by A2,A3,
FUNCT_1:9;
end;

theorem for a,b,c being Element of Funcs(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 Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj(a,x) = Pj((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)
proof
let x be Element of Y;
Pj((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)
=Pj(((a 'or' b) 'or' (c '&' 'not' c)) '&'
(a 'or' 'not' b 'or' c) '&' (a 'or' 'not' b 'or' 'not' c),x)
by BVFUNC_1:14
.=Pj(((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
.=Pj((a 'or' b) '&'
(a 'or' 'not' b 'or' c) '&' (a 'or' 'not' b 'or' 'not' c),x)
by BVFUNC_1:12
.=Pj((a 'or' b) '&'
((a 'or' 'not' b 'or' c) '&' (a 'or' 'not' b 'or' 'not' c)),x)
by BVFUNC_1:7
.=Pj((a 'or' b) '&' ((a 'or' 'not' b) 'or' (c '&' 'not' c)),x)
by BVFUNC_1:14
.=Pj((a 'or' b) '&' ((a 'or' 'not' b) 'or' O_el(Y)),x)
by BVFUNC_4:5
.=Pj((a 'or' b) '&' (a 'or' 'not' b),x)
by BVFUNC_1:12
.=Pj(a 'or' (b '&' 'not' b),x)
by BVFUNC_1:14
.=Pj(a 'or' O_el(Y),x) by BVFUNC_4:5
.=Pj(a,x) by BVFUNC_1:12;
hence thesis;
end;
consider k3 being Function such that
A2: (a)=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: (a 'or' b 'or' c) '&' (a 'or' b 'or' 'not' c) '&' (a 'or' 'not' b 'or' c)
'&' (a 'or' 'not' b 'or' 'not' c)=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence (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) by A2,A3,
FUNCT_1:9;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
a '&' b = a '&' ('not' a 'or' b)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj(a '&' b,x) =
Pj(a '&' ('not' a 'or' b),x)
proof
let x be Element of Y;
Pj(a '&' ('not' a 'or' b),x)
=Pj(a,x) '&' Pj('not' a 'or' b,x)
by VALUAT_1:def 6
.=Pj(a,x) '&' (Pj('not' a,x) 'or' Pj(b,x))
by BVFUNC_1:def 7
.=Pj(a,x) '&' Pj('not' a,x) 'or' Pj(a,x) '&' Pj(b,x)
by BINARITH:22
.=Pj(a,x) '&' 'not' Pj(a,x) 'or' Pj(a,x) '&' Pj(b,x)
by VALUAT_1:def 5
.=FALSE 'or' Pj(a,x) '&' Pj(b,x)
by MARGREL1:46
.=Pj(a,x) '&' Pj(b,x)
by BINARITH:7
.=Pj(a '&' b,x) by VALUAT_1:def 6;
hence thesis;
end;
consider k3 being Function such that
A2: (a '&' b)=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: a '&' ('not' a 'or' b)
=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence thesis by A2,A3,FUNCT_1:9;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
a 'or' b = a 'or' ('not' a '&' b)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj(a 'or' b,x) =
Pj(a 'or' ('not' a '&' b),x)
proof
let x be Element of Y;
Pj(a 'or' ('not' a '&' b),x)
=Pj(a,x) 'or' Pj('not' a '&' b,x)
by BVFUNC_1:def 7
.=Pj(a,x) 'or' (Pj('not' a,x) '&' Pj(b,x))
by VALUAT_1:def 6
.=(Pj(a,x) 'or' Pj('not' a,x)) '&' (Pj(a,x) 'or' Pj(b,x))
by BINARITH:23
.=(Pj(a,x) 'or' 'not' Pj(a,x)) '&' (Pj(a,x) 'or' Pj(b,x))
by VALUAT_1:def 5
.=TRUE '&' (Pj(a,x) 'or' Pj(b,x))
by BINARITH:18
.=Pj(a,x) 'or' Pj(b,x)
by MARGREL1:50
.=Pj(a 'or' b,x) by BVFUNC_1:def 7;
hence thesis;
end;
consider k3 being Function such that
A2: (a 'or' b)=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: a 'or' ('not' a '&' b)
=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence thesis by A2,A3,FUNCT_1:9;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
a 'xor' b = 'not'( a 'eqv' b)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj(a 'xor' b,x) = Pj('not'( a 'eqv' b),x)
proof
let x be Element of Y;
Pj(a 'xor' b,x)
=Pj(('not' a '&' b) 'or' (a '&' 'not' b),x)
by BVFUNC_4:9
.=Pj('not' 'not'( ('not' a '&' b) 'or' (a '&' 'not' b)),x)
by BVFUNC_1:4
.=Pj('not'('not'('not' a '&' b) '&' 'not'( a '&' 'not' b)),x)
by BVFUNC_1:16
.=Pj('not'( ('not' 'not' a 'or' 'not' b) '&' 'not'( a '&' 'not' b)),x)
by BVFUNC_1:17
.=Pj('not'( ('not' 'not' a 'or' 'not' b) '&' ('not' a 'or' 'not' 'not' b)),x)
by BVFUNC_1:17
.=Pj('not'( (a 'or' 'not' b) '&' ('not' a 'or' 'not' 'not' b)),x)
by BVFUNC_1:4
.=Pj('not'( (a 'or' 'not' b) '&' ('not' a 'or' b)),x)
by BVFUNC_1:4
.=Pj('not'( (b 'imp' a) '&' ('not' a 'or' b)),x)
by BVFUNC_4:8
.=Pj('not'( (b 'imp' a) '&' (a 'imp' b)),x)
by BVFUNC_4:8
.=Pj('not'( a 'eqv' b),x) by BVFUNC_4:7;
hence thesis;
end;
consider k3 being Function such that
A2: (a 'xor' b)=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: 'not'( a 'eqv' b)
=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence thesis by A2,A3,FUNCT_1:9;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
a 'xor' b = (a 'or' b) '&' ('not' a 'or' 'not' b)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj(a 'xor' b,x) = Pj((a 'or' b) '&' ('not' a 'or' 'not' b),x)
proof
let x be Element of Y;
Pj((a 'or' b) '&' ('not' a 'or' 'not' b),x)
=Pj(a 'or' b,x) '&' Pj('not' a 'or' 'not' b,x) by VALUAT_1:def 6
.=(Pj(a,x) 'or' Pj(b,x)) '&' Pj('not' a 'or' 'not' b,x) by BVFUNC_1:def 7
.=(Pj(a,x) 'or' Pj(b,x)) '&' (Pj('not' a,x) 'or' Pj('not'
b,x)) by BVFUNC_1:def 7
.=(Pj('not' a,x) '&' (Pj(a,x) 'or' Pj(b,x))) 'or'
((Pj(a,x) 'or' Pj(b,x)) '&' Pj('not' b,x)) by BINARITH:22
.=(Pj('not' a,x) '&' Pj(a,x) 'or' Pj('not' a,x) '&' Pj(b,x)) 'or'
(Pj('not' b,x) '&' (Pj(a,x) 'or' Pj(b,x)))
by BINARITH:22
.=(Pj('not' a,x) '&' Pj(a,x) 'or' Pj('not' a,x) '&' Pj(b,x)) 'or'
(Pj('not' b,x) '&' Pj(a,x) 'or' Pj('not' b,x) '&' Pj(b,x))
by BINARITH:22
.=('not' Pj(a,x) '&' Pj(a,x) 'or' Pj('not' a,x) '&' Pj(b,x)) 'or'
(Pj('not' b,x) '&' Pj(a,x) 'or' Pj('not' b,x) '&' Pj(b,x))
by VALUAT_1:def 5
.=('not' Pj(a,x) '&' Pj(a,x) 'or' Pj('not' a,x) '&' Pj(b,x)) 'or'
(Pj('not' b,x) '&' Pj(a,x) 'or' 'not' Pj(b,x) '&' Pj(b,x))
by VALUAT_1:def 5
.=(FALSE 'or' Pj('not' a,x) '&' Pj(b,x)) 'or'
(Pj('not' b,x) '&' Pj(a,x) 'or' 'not' Pj(b,x) '&' Pj(b,x))
by MARGREL1:46
.=(FALSE 'or' Pj('not' a,x) '&' Pj(b,x)) 'or'
(Pj('not' b,x) '&' Pj(a,x) 'or' FALSE)
by MARGREL1:46
.=(Pj('not' a,x) '&' Pj(b,x)) 'or'
(Pj('not' b,x) '&' Pj(a,x) 'or' FALSE)
by BINARITH:7
.=(Pj('not' a,x) '&' Pj(b,x)) 'or' (Pj(a,x) '&' Pj('not' b,x)) by BINARITH:7
.=Pj('not' a '&' b,x) 'or' (Pj(a,x) '&' Pj('not' b,x))
by VALUAT_1:def 6
.=Pj('not' a '&' b,x) 'or' Pj(a '&' 'not' b,x)
by VALUAT_1:def 6
.=Pj('not' a '&' b 'or' a '&' 'not' b,x)
by BVFUNC_1:def 7
.=Pj(a 'xor' b,x) by BVFUNC_4:9;
hence thesis;
end;
consider k3 being Function such that
A2: (a 'xor' b)=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: (a 'or' b) '&' ('not' a 'or' 'not' b)
=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence thesis by A2,A3,FUNCT_1:9;
end;

theorem for a being Element of Funcs(Y,BOOLEAN) holds
a 'xor' I_el(Y) = 'not' a
proof
let a be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj(a 'xor' I_el(Y),x) = Pj('not' a,x)
proof
let x be Element of Y;
Pj(a 'xor' I_el(Y),x)
=Pj(('not' a '&' I_el(Y)) 'or' (a '&' 'not' I_el(Y)),x) by BVFUNC_4:9
.=Pj(('not' a '&' I_el(Y)) 'or' (a '&' O_el(Y)),x) by BVFUNC_1:5
.=Pj(('not' a '&' I_el(Y)) 'or' O_el(Y),x) by BVFUNC_1:8
.=Pj('not' a '&' I_el(Y),x) by BVFUNC_1:12
.=Pj('not' a,x) by BVFUNC_1:9;
hence thesis;
end;
consider k3 being Function such that
A2: (a 'xor' I_el(Y))=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: 'not' a
=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence thesis by A2,A3,FUNCT_1:9;
end;

theorem for a being Element of Funcs(Y,BOOLEAN) holds
a 'xor' O_el(Y) = a
proof
let a be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj(a 'xor' O_el(Y),x) = Pj(a,x)
proof
let x be Element of Y;
Pj(a 'xor' O_el(Y),x)
=Pj(('not' a '&' O_el(Y)) 'or' (a '&' 'not' O_el(Y)),x) by BVFUNC_4:9
.=Pj(('not' a '&' O_el(Y)) 'or' (a '&' I_el(Y)),x) by BVFUNC_1:5
.=Pj(('not' a '&' O_el(Y)) 'or' a,x) by BVFUNC_1:9
.=Pj(O_el(Y) 'or' a,x) by BVFUNC_1:8
.=Pj(a,x) by BVFUNC_1:12;
hence thesis;
end;
consider k3 being Function such that
A2: (a 'xor' O_el(Y))=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: a
=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence thesis by A2,A3,FUNCT_1:9;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
a 'xor' b = 'not' a 'xor' 'not' b
proof
let a,b be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj(a 'xor' b,x) = Pj('not' a 'xor' 'not' b,x)
proof
let x be Element of Y;
Pj('not' a 'xor' 'not' b,x)
=Pj(('not' 'not' a '&' 'not' b) 'or' ('not' a '&' 'not' 'not' b),x)
by BVFUNC_4:9
.=Pj((a '&' 'not' b) 'or' ('not' a '&' 'not' 'not' b),x) by BVFUNC_1:4
.=Pj((a '&' 'not' b) 'or' ('not' a '&' b),x) by BVFUNC_1:4
.=Pj(a 'xor' b,x) by BVFUNC_4:9;
hence thesis;
end;
consider k3 being Function such that
A2: (a 'xor' b)=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: 'not' a 'xor' 'not' b
=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence thesis by A2,A3,FUNCT_1:9;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
'not'( a 'xor' b) = a 'xor' 'not' b
proof
let a,b be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj('not'( a 'xor' b),x) = Pj(a 'xor' 'not' b,x)
proof
let x be Element of Y;
Pj('not'( a 'xor' b),x)
=Pj('not'( ('not' a '&' b) 'or' (a '&' 'not' b)),x) by BVFUNC_4:9
.=Pj(('not'('not' a '&' b) '&' 'not'( a '&' 'not' b)),x) by BVFUNC_1:16
.=Pj((('not' 'not' a 'or' 'not' b) '&' 'not'( a '&' 'not' b)),x)
by BVFUNC_1:17
.=Pj((('not' 'not' a 'or' 'not' b) '&' ('not' a 'or' 'not' 'not'
b)),x) by BVFUNC_1:17
.=Pj(((a 'or' 'not' b) '&' ('not' a 'or' 'not' 'not' b)),x) by BVFUNC_1:4
.=Pj(((a 'or' 'not' b) '&' ('not' a 'or' b)),x) by BVFUNC_1:4
.=Pj(((a 'or' 'not' b) '&' 'not' a 'or' (a 'or' 'not'
b) '&' b),x) by BVFUNC_1:15
.=Pj(((a '&' 'not' a 'or' 'not' b '&' 'not' a) 'or' (a 'or' 'not'
b) '&' b),x) by BVFUNC_1:15
.=Pj(((O_el(Y) 'or' 'not' b '&' 'not' a) 'or' (a 'or' 'not'
b) '&' b),x) by BVFUNC_4:5
.=Pj((('not' b '&' 'not' a) 'or' (a 'or' 'not' b) '&' b),x) by BVFUNC_1:12
.=Pj((('not' b '&' 'not' a) 'or' (a '&' b 'or' 'not' b '&' b)),x)
by BVFUNC_1:15
.=Pj((('not' b '&' 'not' a) 'or' (a '&' b 'or' O_el(Y))),x) by BVFUNC_4:5
.=Pj(('not' b '&' 'not' a) 'or' (a '&' b),x) by BVFUNC_1:12
.=Pj(('not' a '&' 'not' b) 'or' (a '&' 'not' 'not' b),x) by BVFUNC_1:4
.=Pj(a 'xor' 'not' b,x) by BVFUNC_4:9;
hence thesis;
end;
consider k3 being Function such that
A2: ('not'( a 'xor' b))=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: a 'xor' 'not' b
=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence thesis by A2,A3,FUNCT_1:9;
end;

theorem Th18: for a,b being Element of Funcs(Y,BOOLEAN) holds
a 'eqv' b = (a 'or' 'not' b) '&' ('not' a 'or' b)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj(a 'eqv' b,x) = Pj((a 'or' 'not' b) '&' ('not' a 'or' b),x)
proof
let x be Element of Y;
Pj((a 'or' 'not' b) '&' ('not' a 'or' b),x)
=Pj((a 'or' 'not' b) '&' (a 'imp' b),x) by BVFUNC_4:8
.=Pj((a 'imp' b) '&' (b 'imp' a),x) by BVFUNC_4:8
.=Pj(a 'eqv' b,x) by BVFUNC_4:7;
hence thesis;
end;
consider k3 being Function such that
A2: (a 'eqv' b)=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: (a 'or' 'not' b) '&' ('not' a 'or' b)
=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence thesis by A2,A3,FUNCT_1:9;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
a 'eqv' b = (a '&' b) 'or' ('not' a '&' 'not' b)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj(a 'eqv' b,x) = Pj((a '&' b) 'or' ('not' a '&' 'not' b),x)
proof
let x be Element of Y;
Pj((a '&' b) 'or' ('not' a '&' 'not' b),x)
=Pj(((a '&' b) 'or' 'not' a) '&' ((a '&' b) 'or' 'not' b),x)
by BVFUNC_1:14
.=Pj(((a 'or' 'not' a) '&' (b 'or' 'not' a)) '&' ((a '&' b) 'or' 'not' b),x)
by BVFUNC_1:14
.=Pj(((a 'or' 'not' a) '&' (b 'or' 'not' a)) '&'
((a 'or' 'not' b) '&' (b 'or' 'not' b)),x)
by BVFUNC_1:14
.=Pj((I_el(Y) '&' (b 'or' 'not' a)) '&'
((a 'or' 'not' b) '&' (b 'or' 'not' b)),x)
by BVFUNC_4:6
.=Pj((I_el(Y) '&' (b 'or' 'not' a)) '&'
((a 'or' 'not' b) '&' I_el(Y)),x)
by BVFUNC_4:6
.=Pj((b 'or' 'not' a) '&'
((a 'or' 'not' b) '&' I_el(Y)),x)
by BVFUNC_1:9
.=Pj((b 'or' 'not' a) '&' (a 'or' 'not' b),x)
by BVFUNC_1:9
.=Pj(a 'eqv' b,x) by Th18;
hence thesis;
end;
consider k3 being Function such that
A2: (a 'eqv' b)=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: (a '&' b) 'or' ('not' a '&' 'not' b)
=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence thesis by A2,A3,FUNCT_1:9;
end;

theorem for a being Element of Funcs(Y,BOOLEAN) holds
a 'eqv' I_el(Y) = a
proof
let a be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj(a 'eqv' I_el(Y),x) = Pj(a,x)
proof
let x be Element of Y;
Pj(a 'eqv' I_el(Y),x)
=Pj((a 'imp' I_el(Y)) '&' (I_el(Y) 'imp' a),x)
by BVFUNC_4:7
.=Pj(('not' a 'or' I_el(Y)) '&' (I_el(Y) 'imp' a),x)
by BVFUNC_4:8
.=Pj(('not' a 'or' I_el(Y)) '&' ('not' I_el(Y) 'or' a),x)
by BVFUNC_4:8
.=Pj(I_el(Y) '&' ('not' I_el(Y) 'or' a),x)
by BVFUNC_1:13
.=Pj(I_el(Y) '&' (O_el(Y) 'or' a),x)
by BVFUNC_1:5
.=Pj(I_el(Y) '&' a,x)
by BVFUNC_1:12
.=Pj(a,x) by BVFUNC_1:9;
hence thesis;
end;
consider k3 being Function such that
A2: (a 'eqv' I_el(Y))=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: a
=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence thesis by A2,A3,FUNCT_1:9;
end;

theorem for a being Element of Funcs(Y,BOOLEAN) holds
a 'eqv' O_el(Y) = 'not' a
proof
let a be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj(a 'eqv' O_el(Y),x) = Pj('not' a,x)
proof
let x be Element of Y;
Pj(a 'eqv' O_el(Y),x)
=Pj((a 'imp' O_el(Y)) '&' (O_el(Y) 'imp' a),x)
by BVFUNC_4:7
.=Pj(('not' a 'or' O_el(Y)) '&' (O_el(Y) 'imp' a),x)
by BVFUNC_4:8
.=Pj(('not' a 'or' O_el(Y)) '&' ('not' O_el(Y) 'or' a),x)
by BVFUNC_4:8
.=Pj('not' a '&' ('not' O_el(Y) 'or' a),x)
by BVFUNC_1:12
.=Pj('not' a '&' (I_el(Y) 'or' a),x)
by BVFUNC_1:5
.=Pj('not' a '&' I_el(Y),x)
by BVFUNC_1:13
.=Pj('not' a,x) by BVFUNC_1:9;
hence thesis;
end;
consider k3 being Function such that
A2: (a 'eqv' O_el(Y))=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: 'not' a
=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence thesis by A2,A3,FUNCT_1:9;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
'not'( a 'eqv' b) = (a 'eqv' 'not' b)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
A1:for x being Element of Y holds
Pj('not'( a 'eqv' b),x) = Pj(a 'eqv' 'not' b,x)
proof
let x be Element of Y;
Pj('not'( a 'eqv' b),x)
=Pj('not'( (a 'imp' b) '&' (b 'imp' a)),x) by BVFUNC_4:7
.=Pj('not'( ('not' a 'or' b) '&' (b 'imp' a)),x) by BVFUNC_4:8
.=Pj('not'( ('not' a 'or' b) '&' ('not' b 'or' a)),x) by BVFUNC_4:8
.=Pj('not'('not' a 'or' b) 'or' 'not'('not' b 'or' a),x) by BVFUNC_1:17
.=Pj(('not' 'not' a '&' 'not' b) 'or' 'not'('not' b 'or' a),x) by BVFUNC_1:16
.=Pj(('not' 'not' a '&' 'not' b) 'or' ('not' 'not' b '&' 'not' a),x)
by BVFUNC_1:16
.=Pj((a '&' 'not' b) 'or' ('not' 'not' b '&' 'not' a),x) by BVFUNC_1:4
.=Pj((a '&' 'not' b) 'or' (b '&' 'not' a),x) by BVFUNC_1:4
.=Pj(((a '&' 'not' b) 'or' b) '&' ((a '&' 'not' b) 'or' 'not'
a),x) by BVFUNC_1:14
.=Pj(((a 'or' b) '&' ('not' b 'or' b)) '&' ((a '&' 'not' b) 'or' 'not' a),x)
by BVFUNC_1:14
.=Pj(((a 'or' b) '&' ('not' b 'or' b)) '&'
((a 'or' 'not' a) '&' ('not' b 'or' 'not' a)),x) by BVFUNC_1:14
.=Pj(((a 'or' b) '&' I_el(Y)) '&'
((a 'or' 'not' a) '&' ('not' b 'or' 'not' a)),x) by BVFUNC_4:6
.=Pj(((a 'or' b) '&' I_el(Y)) '&'
(I_el(Y) '&' ('not' b 'or' 'not' a)),x) by BVFUNC_4:6
.=Pj((a 'or' b) '&' (I_el(Y) '&' ('not' b 'or' 'not' a)),x) by BVFUNC_1:9
.=Pj((a 'or' b) '&' ('not' b 'or' 'not' a),x) by BVFUNC_1:9
.=Pj(('not' a 'or' 'not' b) '&' ('not' 'not' b 'or' a),x) by BVFUNC_1:4
.=Pj(('not' a 'or' 'not' b) '&' ('not' b 'imp' a),x) by BVFUNC_4:8
.=Pj((a 'imp' 'not' b) '&' ('not' b 'imp' a),x) by BVFUNC_4:8
.=Pj(a 'eqv' 'not' b,x) by BVFUNC_4:7;
hence thesis;
end;
consider k3 being Function such that
A2: ('not'( a 'eqv' b))=k3 & dom k3=Y & rng k3 c= BOOLEAN
by FUNCT_2:def 2;
consider k4 being Function such that
A3: (a 'eqv' 'not' b)
=k4 & dom k4=Y & rng k4 c= BOOLEAN
by FUNCT_2:def 2;
Y=dom k3 & Y=dom k4 & (for u being set
st u in Y holds k3.u=k4.u)by A1,A2,A3;
hence thesis by A2,A3,FUNCT_1:9;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
'not' a '<' (a 'imp' b) 'eqv' 'not' a
proof
let a,b be Element of Funcs(Y,BOOLEAN);
let z be Element of Y;
assume A1:Pj('not' a,z)=TRUE;
then 'not' Pj(a,z)=TRUE by VALUAT_1:def 5;
then A2:Pj(a,z)=FALSE by MARGREL1:41;
Pj((a 'imp' b) 'eqv' 'not' a,z)
=Pj(('not' a 'or' b) 'eqv' 'not' a,z) by BVFUNC_4:8
.=Pj((('not' a 'or' b) 'imp' 'not' a) '&' ('not' a 'imp' ('not'
a 'or' b)),z) by BVFUNC_4:7
.=Pj(('not'('not' a 'or' b) 'or' 'not' a) '&' ('not' a 'imp' ('not'
a 'or' b)),z) by BVFUNC_4:8
.=Pj(('not'('not' a 'or' b) 'or' 'not' a) '&' ('not' 'not' a 'or' ('not'
a 'or' b)),z) by BVFUNC_4:8
.=Pj('not'('not' a 'or' b) 'or' 'not' a,z) '&' Pj('not' 'not' a 'or' ('not'
a 'or' b),z)
by VALUAT_1:def 6
.=(Pj('not'('not' a 'or' b),z) 'or' Pj('not' a,z)) '&'
Pj('not' 'not' a 'or' ('not' a 'or' b),z)
by BVFUNC_1:def 7
.=('not' Pj('not' a 'or' b,z) 'or' Pj('not' a,z)) '&'
Pj('not' 'not' a 'or' ('not'
a 'or' b),z)
by VALUAT_1:def 5
.=('not'( Pj('not' a,z) 'or' Pj(b,z)) 'or' Pj('not' a,z)) '&' Pj('not' 'not'
a 'or' ('not' a 'or' b),z)
by BVFUNC_1:def 7
.=('not'('not' Pj(a,z) 'or' Pj(b,z)) 'or' Pj('not' a,z)) '&' Pj('not' 'not'
a 'or' ('not' a 'or' b),z)
by VALUAT_1:def 5
.=(('not' 'not' Pj(a,z) '&' 'not' Pj(b,z)) 'or' Pj('not' a,z)) '&'
Pj('not' 'not'
a 'or' ('not' a 'or' b),z)
by BINARITH:10
.=((Pj(a,z) '&' 'not' Pj(b,z)) 'or' Pj('not' a,z)) '&'
Pj('not' 'not' a 'or' (
'not' a 'or' b),z)
by MARGREL1:40
.=((Pj(a,z) '&' 'not' Pj(b,z)) 'or' Pj('not' a,z)) '&'
(Pj('not' 'not' a,z) 'or' Pj('not' a 'or' b,z))
by BVFUNC_1:def 7
.=((Pj(a,z) '&' 'not' Pj(b,z)) 'or' Pj('not' a,z)) '&'
(Pj('not' 'not' a,z) 'or' (Pj('not' a,z) 'or' Pj(b,z)))
by BVFUNC_1:def 7
.=((Pj(a,z) '&' 'not' Pj(b,z)) 'or' Pj('not' a,z)) '&'
(Pj('not' 'not' a,z) 'or' ('not' Pj(a,z) 'or' Pj(b,z)))
by VALUAT_1:def 5
.=((Pj(a,z) '&' 'not' Pj(b,z)) 'or' Pj('not' a,z)) '&'
(Pj(a,z) 'or' ('not' Pj(a,z) 'or' Pj(b,z)))
by BVFUNC_1:4
.=((Pj(a,z) '&' 'not' Pj(b,z)) 'or' Pj('not' a,z)) '&'
(Pj(a,z) 'or' (Pj('not' a,z) 'or' Pj(b,z)))
by VALUAT_1:def 5
.=TRUE '&' (FALSE 'or' (TRUE 'or' Pj(b,z)))
by A1,A2,BINARITH:19
.=FALSE 'or' (TRUE 'or' Pj(b,z))
by MARGREL1:50
.=(TRUE 'or' Pj(b,z))
by BINARITH:7
.=TRUE by BINARITH:19;
hence thesis;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
'not' a '<' (b 'imp' a) 'eqv' 'not' b
proof
let a,b be Element of Funcs(Y,BOOLEAN);
let z be Element of Y;
assume Pj('not' a,z)=TRUE;
then A1: 'not' Pj(a,z)=TRUE by VALUAT_1:def 5;
Pj((b 'imp' a) 'eqv' 'not' b,z)
=Pj(('not' b 'or' a) 'eqv' 'not' b,z) by BVFUNC_4:8
.=Pj((('not' b 'or' a) 'imp' 'not' b) '&' ('not' b 'imp' ('not'
b 'or' a)),z) by BVFUNC_4:7
.=Pj(('not'('not' b 'or' a) 'or' 'not' b) '&' ('not' b 'imp' ('not'
b 'or' a)),z) by BVFUNC_4:8
.=Pj(('not'('not' b 'or' a) 'or' 'not' b) '&' ('not' 'not' b 'or' ('not'
b 'or' a)),z) by BVFUNC_4:8
.=Pj('not'('not' b 'or' a) 'or' 'not' b,z) '&' Pj('not' 'not' b 'or' ('not'
b 'or' a),z)
by VALUAT_1:def 6
.=(Pj('not'('not' b 'or' a),z) 'or' Pj('not' b,z)) '&'
Pj('not' 'not' b 'or' ('not' b 'or' a),z)
by BVFUNC_1:def 7
.=('not' Pj('not' b 'or' a,z) 'or' Pj('not' b,z)) '&'
Pj('not' 'not' b 'or' ('not'
b 'or' a),z)
by VALUAT_1:def 5
.=('not'( Pj('not' b,z) 'or' Pj(a,z)) 'or' Pj('not' b,z)) '&' Pj('not' 'not'
b 'or' ('not' b 'or' a),z)
by BVFUNC_1:def 7
.=('not'('not' Pj(b,z) 'or' Pj(a,z)) 'or' Pj('not' b,z)) '&' Pj('not' 'not'
b 'or' ('not' b 'or' a),z)
by VALUAT_1:def 5
.=(('not' 'not' Pj(b,z) '&' 'not' Pj(a,z)) 'or' Pj('not' b,z)) '&'
Pj('not' 'not'
b 'or' ('not' b 'or' a),z)
by BINARITH:10
.=((Pj(b,z) '&' 'not' Pj(a,z)) 'or' Pj('not' b,z)) '&'
Pj('not' 'not' b 'or' (
'not' b 'or' a),z)
by MARGREL1:40
.=((Pj(b,z) '&' 'not' Pj(a,z)) 'or' Pj('not' b,z)) '&' (Pj('not' 'not'
b,z) 'or' Pj('not' b 'or' a,z))
by BVFUNC_1:def 7
.=((Pj(b,z) '&' 'not' Pj(a,z)) 'or' Pj('not' b,z)) '&'
(Pj('not' 'not' b,z) 'or' (Pj('not' b,z) 'or' Pj(a,z))) by BVFUNC_1:def 7
.=((Pj(b,z) '&' 'not' Pj(a,z)) 'or' Pj('not' b,z)) '&'
(Pj('not' 'not' b,z) 'or' ('not' Pj(b,z) 'or' Pj(a,z))) by VALUAT_1:def 5
.=((Pj(b,z) '&' 'not' Pj(a,z)) 'or' Pj('not' b,z)) '&'
(Pj(b,z) 'or' ('not' Pj(b,z) 'or' Pj(a,z))) by BVFUNC_1:4
.=((TRUE '&' Pj(b,z)) 'or' Pj('not' b,z)) '&'
(Pj(b,z) 'or' ('not' Pj(b,z) 'or' FALSE)) by A1,MARGREL1:41
.=(Pj(b,z) 'or' Pj('not' b,z)) '&'
(Pj(b,z) 'or' ('not' Pj(b,z) 'or' FALSE)) by MARGREL1:50
.=(Pj(b,z) 'or' 'not' Pj(b,z)) '&'
(Pj(b,z) 'or' ('not' Pj(b,z) 'or' FALSE)) by VALUAT_1:def 5
.=TRUE '&' (Pj(b,z) 'or' ('not' Pj(b,z) 'or' FALSE)) by BINARITH:18
.=Pj(b,z) 'or' ('not' Pj(b,z) 'or' FALSE) by MARGREL1:50
.=Pj(b,z) 'or' 'not' Pj(b,z) 'or' FALSE by BINARITH:20
.=TRUE 'or' FALSE by BINARITH:18
.=TRUE by BINARITH:19;
hence thesis;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
a '<' (a 'or' b) 'eqv' (b 'or' a) 'eqv' a
proof
let a,b be Element of Funcs(Y,BOOLEAN);
let z be Element of Y;
assume A1:Pj(a,z)=TRUE;
A2:Pj((a 'or' b) 'eqv' (b 'or' a),z)
=Pj(((a 'or' b) 'imp' (a 'or' b)) '&' ((a 'or' b) 'imp' (a 'or' b)),z)
by BVFUNC_4:7
.=Pj((a 'or' b) 'imp' (a 'or' b),z) '&' Pj((a 'or' b) 'imp' (a 'or' b),z)
by VALUAT_1:def 6
.=Pj((a 'or' b) 'imp' (a 'or' b),z)
by BINARITH:16
.=Pj('not'( a 'or' b) 'or' (a 'or' b),z)
by BVFUNC_4:8
.=Pj(I_el(Y),z) by BVFUNC_4:6
.=TRUE by BVFUNC_1:def 14;
Pj((a 'or' b) 'eqv' (b 'or' a) 'eqv' a,z)
=Pj((((a 'or' b) 'eqv' (a 'or' b)) 'imp' a) '&'
(a 'imp' ((a 'or' b) 'eqv' (a 'or' b))),z) by BVFUNC_4:7
.=Pj(((a 'or' b) 'eqv' (a 'or' b)) 'imp' a,z) '&'
Pj(a 'imp' ((a 'or' b) 'eqv' (a 'or' b)),z)
by VALUAT_1:def 6
.=Pj('not'( (a 'or' b) 'eqv' (a 'or' b)) 'or' a,z) '&'
Pj(a 'imp' ((a 'or' b) 'eqv' (a 'or' b)),z)
by BVFUNC_4:8
.=Pj('not'( (a 'or' b) 'eqv' (a 'or' b)) 'or' a,z) '&'
Pj('not' a 'or' ((a 'or' b) 'eqv' (a 'or' b)),z)
by BVFUNC_4:8
.=(Pj('not'( (a 'or' b) 'eqv' (a 'or' b)),z) 'or' Pj(a,z)) '&'
Pj('not' a 'or' ((a 'or' b) 'eqv' (a 'or' b)),z)
by BVFUNC_1:def 7
.=(Pj('not'( (a 'or' b) 'eqv' (a 'or' b)),z) 'or' Pj(a,z)) '&'
(Pj('not' a,z) 'or' Pj((a 'or' b) 'eqv' (a 'or' b),z))
by BVFUNC_1:def 7
.=('not' Pj((a 'or' b) 'eqv' (a 'or' b),z) 'or' Pj(a,z)) '&'
(Pj('not' a,z) 'or' Pj((a 'or' b) 'eqv' (a 'or' b),z)) by VALUAT_1:def 5
.=(FALSE 'or' Pj(a,z)) '&' (Pj('not' a,z) 'or' TRUE) by A2,MARGREL1:41
.=Pj(a,z) '&' (Pj('not' a,z) 'or' TRUE) by BINARITH:7
.=Pj(a,z) '&' TRUE by BINARITH:19
.=TRUE by A1,MARGREL1:50;
hence thesis;
end;

theorem for a being Element of Funcs(Y,BOOLEAN) holds
a 'imp' ('not' a 'eqv' 'not' a) = I_el(Y)
proof
let a be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds
Pj(a 'imp' ('not' a 'eqv' 'not' a),x) = TRUE
proof
let x be Element of Y;
Pj(a 'imp' ('not' a 'eqv' 'not' a),x)
=Pj('not' a 'or' ('not' a 'eqv' 'not' a),x) by BVFUNC_4:8
.=Pj('not' a 'or' (('not' a 'imp' 'not' a) '&' ('not' a 'imp' 'not'
a)),x) by BVFUNC_4:7
.=Pj('not' a 'or' ('not' a 'imp' 'not' a),x) by BVFUNC_1:6
.=Pj('not' a 'or' ('not' 'not' a 'or' 'not' a),x) by BVFUNC_4:8
.=Pj('not' a 'or' I_el(Y),x) by BVFUNC_4:6
.=Pj(I_el(Y),x) by BVFUNC_1:13
.=TRUE by BVFUNC_1:def 14;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b being Element of Funcs(Y,BOOLEAN) holds
((a 'imp' b) 'imp' a) 'imp' a = I_el(Y)
proof
let a,b be Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds
Pj(((a 'imp' b) 'imp' a) 'imp' a,x) = TRUE
proof
let x be Element of Y;
Pj(((a 'imp' b) 'imp' a) 'imp' a,x)
=Pj('not'( (a 'imp' b) 'imp' a) 'or' a,x) by BVFUNC_4:8
.=Pj('not'('not'( a 'imp' b) 'or' a) 'or' a,x) by BVFUNC_4:8
.=Pj('not'('not'('not' a 'or' b) 'or' a) 'or' a,x) by BVFUNC_4:8
.=Pj('not'( ('not' 'not' a '&' 'not' b) 'or' a) 'or' a,x) by BVFUNC_1:16
.=Pj('not'( (a '&' 'not' b) 'or' a) 'or' a,x) by BVFUNC_1:4
.=Pj('not'( ((a 'or' a) '&' ('not' b 'or' a))) 'or' a,x) by BVFUNC_1:14
.=Pj('not'( a '&' ('not' b 'or' a)) 'or' a,x) by BVFUNC_1:10
.=Pj(('not' a 'or' 'not'('not' b 'or' a)) 'or' a,x) by BVFUNC_1:17
.=Pj(('not' a 'or' ('not' 'not' b '&' 'not' a)) 'or' a,x) by BVFUNC_1:16
.=Pj(('not' a 'or' (b '&' 'not' a)) 'or' a,x) by BVFUNC_1:4
.=Pj((('not' a 'or' b) '&' ('not' a 'or' 'not' a)) 'or' a,x) by BVFUNC_1:14
.=Pj((('not' a 'or' b) '&' 'not' a) 'or' a,x) by BVFUNC_1:10
.=Pj((('not' a 'or' b) 'or' a) '&' ('not' a 'or' a),x) by BVFUNC_1:14
.=Pj((('not' a 'or' b) 'or' a) '&' I_el(Y),x) by BVFUNC_4:6
.=Pj(('not' a 'or' b) 'or' a,x) by BVFUNC_1:9
.=Pj(b 'or' ('not' a 'or' a),x) by BVFUNC_1:11
.=Pj(b 'or' I_el(Y),x) by BVFUNC_4:6
.=Pj(I_el(Y),x) by BVFUNC_1:13
.=TRUE by BVFUNC_1:def 14;
hence thesis;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b,c,d being Element of Funcs(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 Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds
Pj(((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_1:17
.=(('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_1:17
.=((('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_1:16
.=(((a '&' 'not' c) 'or' 'not'('not' b 'or' d)) 'or' 'not'('not' c 'or' 'not'
d)) 'or' ('not' a 'or' 'not' b)
by BVFUNC_1:4
.=(((a '&' 'not' c) 'or' ('not' 'not' b '&' 'not' d)) 'or' 'not'('not' c 'or'
'not' d)) 'or' ('not' a 'or' 'not' b)
by BVFUNC_1:16
.=(((a '&' 'not' c) 'or' (b '&' 'not' d)) 'or' 'not'('not' c 'or' 'not'
d)) 'or' ('not' a 'or' 'not' b)
by BVFUNC_1:4
.=(((a '&' 'not' c) 'or' (b '&' 'not' d)) 'or' ('not' 'not' c '&' 'not' 'not'
d)) 'or' ('not' a 'or' 'not' b)
by BVFUNC_1:16
.=(((a '&' 'not' c) 'or' (b '&' 'not' d)) 'or'
(c '&' 'not' 'not' d)) 'or' ('not'
a 'or' 'not' b)
by BVFUNC_1:4
.=(((a '&' 'not' c) 'or' (b '&' 'not' d)) 'or' (c '&' d)) 'or' ('not' a 'or'
'not' b)
by BVFUNC_1:4
.=((a '&' 'not' c) 'or' ((b '&' '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 '&' d))))
'or' ('not' a 'or' 'not' b)
by BVFUNC_1:14
.=((a '&' 'not' c) 'or' ((b 'or' (c '&' d)) '&' (('not' d 'or' c) '&' ('not'
d 'or' d))))
'or' ('not' a 'or' 'not' b)
by BVFUNC_1:14
.=((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_1:9
.=((b 'or' (c '&' d)) '&' ('not' d 'or' c)) 'or'
((a '&' '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_1:14
.=((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'
((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_1:13
.=((b 'or' (c '&' d)) '&' ('not' d 'or' c)) 'or' ('not' c 'or' ('not' a 'or'
'not' b))
by BVFUNC_1:9
.=((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_1:14
.=((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_1:11
.=((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_1:13
.=((b 'or' (c '&' d)) 'or' ('not' c 'or' ('not' a 'or' 'not' b))) '&'
I_el(Y) by BVFUNC_1:13
.=(b 'or' (c '&' d)) 'or' ('not' c 'or' ('not' a 'or' 'not' b)) by BVFUNC_1:9
.=(c '&' d) 'or' (b 'or' ('not' c 'or' ('not' a 'or' 'not' b)))
by BVFUNC_1:11
.=(c '&' d) 'or' ((b 'or' ('not' b 'or' 'not' a)) 'or' 'not' c)
by BVFUNC_1:11
.=(c '&' d) 'or' (((b 'or' 'not' b) 'or' 'not' a) 'or' 'not' c)
by BVFUNC_1:11
.=(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_1:13
.=(c '&' d) 'or' I_el(Y) by BVFUNC_1:13
.=I_el(Y) by BVFUNC_1:13;
hence thesis by BVFUNC_1:def 14;
end;
hence thesis by BVFUNC_1:def 14;
end;

theorem for a,b,c being Element of Funcs(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 Element of Funcs(Y,BOOLEAN);
for x being Element of Y holds
Pj((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_1:16
.=('not' 'not' a '&' 'not' b) 'or' (('not' 'not' a '&' 'not'('not'
b 'or' c)) 'or' ('not' a 'or' c)) by BVFUNC_1:16
.=('not' 'not' a '&' 'not' b) 'or' (('not' 'not' a '&'
('not' 'not' b '&' 'not'
c)) 'or' ('not' a 'or' c)) by BVFUNC_1:16
.=(a '&' 'not' b) 'or' (('not' 'not' a '&' ('not' 'not' b
'&' 'not' c)) 'or' ('not'
a 'or' c)) by BVFUNC_1:4
.=(a '&' 'not' b) 'or' ((a '&' ('not' 'not' b '&' 'not' c)) 'or' ('not'
a 'or' c)) by BVFUNC_1:4
.=(a '&' 'not' b) 'or' ((a '&' (b '&' 'not' c)) 'or' ('not'
a 'or' c)) by BVFUNC_1:4
.=(a '&' 'not' b) 'or'
((a 'or' ('not' a 'or' c)) '&' ((b '&' 'not' c) 'or' ('not'
a 'or' c))) by BVFUNC_1:14
.=(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'
((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_1:13
.=(a '&' 'not' b) 'or'
((b '&' 'not' c) 'or' ('not' a 'or' c)) by BVFUNC_1:9
.=(a '&' 'not' b) 'or'
((b 'or' ('not' a 'or' c)) '&' ('not' c 'or' ('not' a 'or' c)))
by BVFUNC_1:14
.=(a '&' 'not' b) 'or'
((b 'or' ('not' a 'or' c)) '&' (('not' c 'or' c) 'or' 'not' a))
by BVFUNC_1:11
.=(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_1:13
.=(a '&' 'not' b) 'or' (b 'or' ('not' a 'or' c)) by BVFUNC_1:9
.=(a 'or' (b 'or' ('not' a 'or' c))) '&' ('not' b 'or' (b 'or' ('not'
a 'or' c)))
by BVFUNC_1:14
.=(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))) '&' (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_1:13
.=a 'or' (b 'or' ('not' a 'or' c)) by BVFUNC_1:9
.=a 'or' (('not' a 'or' b) 'or' c) by BVFUNC_1:11
.=a 'or' ('not' a 'or' b) 'or' c by BVFUNC_1:11
.=(a 'or' 'not' a) 'or' b 'or' c by BVFUNC_1:11
.=I_el(Y) 'or' b 'or' c by BVFUNC_4:6
.=I_el(Y) 'or' c by BVFUNC_1:13
.=I_el(Y) by BVFUNC_1:13;
hence thesis by BVFUNC_1:def 14;
end;
hence thesis by BVFUNC_1:def 14;
end;
```