:: Strong arithmetic of real numbers
:: by Andrzej Trybulec
::
:: Received January 1, 1989
:: Copyright (c) 1990-2021 Association of Mizar Users

Lm1: for r, s being Real st ( ( r in REAL+ & s in REAL+ & ex x9, y9 being Element of REAL+ st
( r = x9 & s = y9 & x9 <=' y9 ) ) or ( r in [:{0},REAL+:] & s in [:{0},REAL+:] & ex x9, y9 being Element of REAL+ st
( r = [0,x9] & s = [0,y9] & y9 <=' x9 ) ) or ( s in REAL+ & r in [:{0},REAL+:] ) ) holds
r <= s

proof end;

Lm2: for x being Real
for x1, x2 being Element of REAL st x = [*x1,x2*] holds
( x2 = 0 & x = x1 )

proof end;

Lm3: for x9, y9 being Element of REAL
for x, y being Real st x9 = x & y9 = y holds
+ (x9,y9) = x + y

proof end;

Lm4: {} in {{}}
by TARSKI:def 1;

theorem :: AXIOMS:1

for X, Y being Subset of REAL st ( for x, y being Real st x in X & y in Y holds
x <= y ) holds
ex z being Real st
for x, y being Real st x in X & y in Y holds
( x <= z & z <= y )

proof end;

theorem :: AXIOMS:2

for x, y being Real st x in NAT & y in NAT holds
x + y in NAT

proof end;

theorem :: AXIOMS:3

for A being Subset of REAL st 0 in A & ( for x being Real st x in A holds
x + 1 in A ) holds
NAT c= A

proof end;

theorem :: AXIOMS:4

for k being natural Number holds k = { i where i is Nat : i < k }

proof end;