:: AXIOMS semantic presentation

Lemma13: for r, s being real number st r <= s holds
( ( r in REAL+ & s in REAL+ implies ex x', y' being Element of REAL+ st
( r = x' & s = y' & x' <=' y' ) ) & ( r in [:{0},REAL+ :] & s in [:{0},REAL+ :] implies ex x', y' being Element of REAL+ st
( r = [0,x'] & s = [0,y'] & y' <=' x' ) ) & ( ( not r in REAL+ or not s in REAL+ ) & ( not r in [:{0},REAL+ :] or not s in [:{0},REAL+ :] ) implies ( s in REAL+ & r in [:{0},REAL+ :] ) ) )
by XXREAL_0:def 5;

Lemma16: for r, s being real number st ( ( r in REAL+ & s in REAL+ & ex x', y' being Element of REAL+ st
( r = x' & s = y' & x' <=' y' ) ) or ( r in [:{0},REAL+ :] & s in [:{0},REAL+ :] & ex x', y' being Element of REAL+ st
( r = [0,x'] & s = [0,y'] & y' <=' x' ) ) or ( s in REAL+ & r in [:{0},REAL+ :] ) ) holds
r <= s
proof end;

theorem Th1: :: AXIOMS:1
canceled;

theorem Th2: :: AXIOMS:2
canceled;

theorem Th3: :: AXIOMS:3
canceled;

theorem Th4: :: AXIOMS:4
canceled;

theorem Th5: :: AXIOMS:5
canceled;

theorem Th6: :: AXIOMS:6
canceled;

theorem Th7: :: AXIOMS:7
canceled;

theorem Th8: :: AXIOMS:8
canceled;

theorem Th9: :: AXIOMS:9
canceled;

theorem Th10: :: AXIOMS:10
canceled;

theorem Th11: :: AXIOMS:11
canceled;

theorem Th12: :: AXIOMS:12
canceled;

theorem Th13: :: AXIOMS:13
canceled;

theorem Th14: :: AXIOMS:14
canceled;

theorem Th15: :: AXIOMS:15
canceled;

theorem Th16: :: AXIOMS:16
canceled;

theorem Th17: :: AXIOMS:17
canceled;

theorem Th18: :: AXIOMS:18
canceled;

theorem Th19: :: AXIOMS:19
for x being real number ex y being real number st x + y = 0
proof end;

theorem Th20: :: AXIOMS:20
for x being real number st x <> 0 holds
ex y being real number st x * y = 1
proof end;

Lemma23: for x, y being real number st x <= y & y <= x holds
x = y
by XXREAL_0:1;

Lemma24: for x being real number
for x1, x2 being Element of REAL st x = [*x1,x2*] holds
( x2 = 0 & x = x1 )
proof end;

Lemma28: for x', y' being Element of REAL
for x, y being real number st x' = x & y' = y holds
+ x',y' = x + y
proof end;

Lemma34: {} in {{} }
by TARSKI:def 1;

reconsider o = 0 as Element of REAL+ by ARYTM_2:21;

theorem Th21: :: AXIOMS:21
canceled;

theorem Th22: :: AXIOMS:22
canceled;

theorem Th23: :: AXIOMS:23
canceled;

theorem Th24: :: AXIOMS:24
canceled;

theorem Th25: :: AXIOMS:25
canceled;

theorem Th26: :: AXIOMS:26
for X, Y being Subset of REAL st ( for x, y being real number st x in X & y in Y holds
x <= y ) holds
ex z being real number st
for x, y being real number st x in X & y in Y holds
( x <= z & z <= y )
proof end;

theorem Th27: :: AXIOMS:27
canceled;

theorem Th28: :: AXIOMS:28
for x, y being real number st x in NAT & y in NAT holds
x + y in NAT
proof end;

theorem Th29: :: AXIOMS:29
for A being Subset of REAL st 0 in A & ( for x being real number st x in A holds
x + 1 in A ) holds
NAT c= A
proof end;

theorem Th30: :: AXIOMS:30
for k being natural number holds k = { i where i is Element of NAT : i < k }
proof end;