:: AXIOMS semantic presentation
Lm1:
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;
Lm2:
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
theorem :: AXIOMS:1
canceled;
theorem :: AXIOMS:2
canceled;
theorem :: AXIOMS:3
canceled;
theorem :: AXIOMS:4
canceled;
theorem :: AXIOMS:5
canceled;
theorem :: AXIOMS:6
canceled;
theorem :: AXIOMS:7
canceled;
theorem :: AXIOMS:8
canceled;
theorem :: AXIOMS:9
canceled;
theorem :: AXIOMS:10
canceled;
theorem :: AXIOMS:11
canceled;
theorem :: AXIOMS:12
canceled;
theorem :: AXIOMS:13
canceled;
theorem :: AXIOMS:14
canceled;
theorem :: AXIOMS:15
canceled;
theorem :: AXIOMS:16
canceled;
theorem :: AXIOMS:17
canceled;
theorem :: AXIOMS:18
canceled;
theorem :: AXIOMS:19
theorem :: AXIOMS:20
Lm3:
for x, y being real number st x <= y & y <= x holds
x = y
by XXREAL_0:1;
Lm4:
for x being real number
for x1, x2 being Element of REAL st x = [*x1,x2*] holds
( x2 = 0 & x = x1 )
Lm5:
for x', y' being Element of REAL
for x, y being real number st x' = x & y' = y holds
+ x',y' = x + y
Lm6:
{} in {{} }
by TARSKI:def 1;
reconsider o = 0 as Element of REAL+ by ARYTM_2:21;
theorem :: AXIOMS:21
canceled;
theorem :: AXIOMS:22
canceled;
theorem :: AXIOMS:23
canceled;
theorem :: AXIOMS:24
canceled;
theorem :: AXIOMS:25
canceled;
theorem :: AXIOMS:26
theorem :: AXIOMS:27
canceled;
theorem :: AXIOMS:28
theorem :: AXIOMS:29
theorem :: AXIOMS:30