:: AXIOMS semantic presentation

Lemma1: for b1, b2 being real number holds
( b1 <= b2 implies ( not ( b1 in REAL+ & b2 in REAL+ & ( for b3, b4 being Element of REAL+ holds
not ( b1 = b3 & b2 = b4 & b3 <=' b4 ) ) ) & not ( b1 in [:{0},REAL+ :] & b2 in [:{0},REAL+ :] & ( for b3, b4 being Element of REAL+ holds
not ( b1 = [0,b3] & b2 = [0,b4] & b4 <=' b3 ) ) ) & ( not ( b1 in REAL+ & b2 in REAL+ ) & not ( b1 in [:{0},REAL+ :] & b2 in [:{0},REAL+ :] ) implies ( b2 in REAL+ & b1 in [:{0},REAL+ :] ) ) ) )
by XXREAL_0:def 5;

Lemma2: for b1, b2 being real number holds
( not ( not ( b1 in REAL+ & b2 in REAL+ & ex b3, b4 being Element of REAL+ st
( b1 = b3 & b2 = b4 & b3 <=' b4 ) ) & not ( b1 in [:{0},REAL+ :] & b2 in [:{0},REAL+ :] & ex b3, b4 being Element of REAL+ st
( b1 = [0,b3] & b2 = [0,b4] & b4 <=' b3 ) ) & not ( b2 in REAL+ & b1 in [:{0},REAL+ :] ) ) implies b1 <= b2 )
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 b1 being real number holds
ex b2 being real number st b1 + b2 = 0
proof end;

theorem Th20: :: AXIOMS:20
for b1 being real number holds
not ( b1 <> 0 & ( for b2 being real number holds
not b1 * b2 = 1 ) )
proof end;

Lemma3: for b1, b2 being real number holds
( b1 <= b2 & b2 <= b1 implies b1 = b2 )
by XXREAL_0:1;

Lemma4: for b1 being real number
for b2, b3 being Element of REAL holds
( b1 = [*b2,b3*] implies ( b3 = 0 & b1 = b2 ) )
proof end;

Lemma5: for b1, b2 being Element of REAL
for b3, b4 being real number holds
( b1 = b3 & b2 = b4 implies + b1,b2 = b3 + b4 )
proof end;

Lemma6: {} in {{} }
by TARSKI:def 1;

reconsider c1 = 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 b1, b2 being Subset of REAL holds
not ( ( for b3, b4 being real number holds
( b3 in b1 & b4 in b2 implies b3 <= b4 ) ) & ( for b3 being real number holds
ex b4, b5 being real number st
( b4 in b1 & b5 in b2 & not ( b4 <= b3 & b3 <= b5 ) ) ) )
proof end;

theorem Th27: :: AXIOMS:27
canceled;

theorem Th28: :: AXIOMS:28
for b1, b2 being real number holds
( b1 in NAT & b2 in NAT implies b1 + b2 in NAT )
proof end;

Lemma7: 1 = succ 0
;

theorem Th29: :: AXIOMS:29
for b1 being Subset of REAL holds
( 0 in b1 & ( for b2 being real number holds
( b2 in b1 implies b2 + 1 in b1 ) ) implies NAT c= b1 )
proof end;

theorem Th30: :: AXIOMS:30
for b1 being natural number holds b1 = { b2 where B is Element of NAT : b2 < b1 }
proof end;