:: Strong arithmetic of real numbers :: by Andrzej Trybulec :: :: Received January 1, 1989 :: Copyright (c) 1990-2016 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 ARYTM_2, SUBSET_1, ARYTM_3, ZFMISC_1, CARD_1, XXREAL_0, NUMBERS, ARYTM_0, FUNCOP_1, XBOOLE_0, TARSKI, NAT_1, REAL_1, ORDINAL1; notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, FUNCT_4, ARYTM_2, NUMBERS, ARYTM_0, XCMPLX_0, XREAL_0, XXREAL_0; constructors FUNCT_4, ARYTM_1, ARYTM_0, XCMPLX_0, XXREAL_0, XREAL_0; registrations ORDINAL1, ARYTM_2, XREAL_0; requirements SUBSET, BOOLE, NUMERALS, ARITHM; begin reserve r,s for Real; reserve x,y,z for Real; reserve r,r1,r2 for Element of REAL+; theorem :: AXIOMS:1 for X,Y being Subset of REAL st for x,y st x in X & y in Y holds x <= y ex z st for x,y st x in X & y in Y holds x <= z & z <= y; theorem :: AXIOMS:2 x in NAT & y in NAT implies x + y in NAT; theorem :: AXIOMS:3 for A being Subset of REAL st 0 in A & for x st x in A holds x + 1 in A holds NAT c= A; theorem :: AXIOMS:4 for k being natural Number holds k = { i where i is Nat: i < k };