:: AXIOMS semantic presentation
Lemma1:
for b1, b2 being real number st b1 <= b2 holds
( ( b1 in REAL+ & b2 in REAL+ implies ex b3, b4 being Element of REAL+ st
( b1 = b3 & b2 = b4 & b3 <=' b4 ) ) & ( b1 in [:{0},REAL+ :] & b2 in [:{0},REAL+ :] implies ex b3, b4 being Element of REAL+ st
( b1 = [0,b3] & b2 = [0,b4] & b4 <=' b3 ) ) & ( ( not b1 in REAL+ or not b2 in REAL+ ) & ( not b1 in [:{0},REAL+ :] or not b2 in [:{0},REAL+ :] ) implies ( b2 in REAL+ & b1 in [:{0},REAL+ :] ) ) )
by XXREAL_0:def 5;
Lemma2:
for b1, b2 being real number st ( ( b1 in REAL+ & b2 in REAL+ & ex b3, b4 being Element of REAL+ st
( b1 = b3 & b2 = b4 & b3 <=' b4 ) ) or ( b1 in [:{0},REAL+ :] & b2 in [:{0},REAL+ :] & ex b3, b4 being Element of REAL+ st
( b1 = [0,b3] & b2 = [0,b4] & b4 <=' b3 ) ) or ( b2 in REAL+ & b1 in [:{0},REAL+ :] ) ) holds
b1 <= b2
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
theorem Th20: :: AXIOMS:20
Lemma3:
for b1, b2 being real number st b1 <= b2 & b2 <= b1 holds
b1 = b2
by XXREAL_0:1;
Lemma4:
for b1 being real number
for b2, b3 being Element of REAL st b1 = [*b2,b3*] holds
( b3 = 0 & b1 = b2 )
Lemma5:
for b1, b2 being Element of REAL
for b3, b4 being real number st b1 = b3 & b2 = b4 holds
+ b1,b2 = b3 + b4
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
theorem Th27: :: AXIOMS:27
canceled;
theorem Th28: :: AXIOMS:28
Lemma7:
1 = succ 0
;
theorem Th29: :: AXIOMS:29
theorem Th30: :: AXIOMS:30