:: EXTREAL1 semantic presentation
theorem Th1: :: EXTREAL1:1
theorem Th2: :: EXTREAL1:2
canceled;
theorem Th3: :: EXTREAL1:3
canceled;
theorem Th4: :: EXTREAL1:4
theorem Th5: :: EXTREAL1:5
theorem Th6: :: EXTREAL1:6
canceled;
theorem Th7: :: EXTREAL1:7
Lemma21:
for x being R_eal st x in REAL holds
( x + -infty = -infty & x + +infty = +infty )
by SUPINF_1:1, SUPINF_1:6, SUPINF_2:def 2;
Lemma22:
for x, y being R_eal st x in REAL & y in REAL holds
x + y in REAL
theorem Th8: :: EXTREAL1:8
theorem Th9: :: EXTREAL1:9
theorem Th10: :: EXTREAL1:10
canceled;
theorem Th11: :: EXTREAL1:11
:: deftheorem Def1 defines * EXTREAL1:def 1 :
theorem Th12: :: EXTREAL1:12
canceled;
theorem Th13: :: EXTREAL1:13
for
x,
y being
R_eal for
a,
b being
Real st
x = a &
y = b holds
x * y = a * b
Lemma52:
for x being R_eal
for a being Real st x = a & 0 < a holds
0. < x
by SUPINF_2:def 1;
Lemma53:
for x being R_eal
for a being Real st x = a & a < 0 holds
x < 0.
by SUPINF_2:def 1;
theorem Th14: :: EXTREAL1:14
theorem Th15: :: EXTREAL1:15
theorem Th16: :: EXTREAL1:16
theorem Th17: :: EXTREAL1:17
for
x,
y being
R_eal holds
x * y = y * x
theorem Th18: :: EXTREAL1:18
theorem Th19: :: EXTREAL1:19
theorem Th20: :: EXTREAL1:20
theorem Th21: :: EXTREAL1:21
theorem Th22: :: EXTREAL1:22
theorem Th23: :: EXTREAL1:23
for
x,
y,
z being
R_eal holds
(x * y) * z = x * (y * z)
theorem Th24: :: EXTREAL1:24
theorem Th25: :: EXTREAL1:25
theorem Th26: :: EXTREAL1:26
for
x,
y being
R_eal holds
(
- (x * y) = x * (- y) &
- (x * y) = (- x) * y )
theorem Th27: :: EXTREAL1:27
theorem Th28: :: EXTREAL1:28
Lemma125:
for x, y, z being R_eal st x <> +infty & x <> -infty holds
x * (y + z) = (x * y) + (x * z)
theorem Th29: :: EXTREAL1:29
theorem Th30: :: EXTREAL1:30
:: deftheorem Def2 defines / EXTREAL1:def 2 :
theorem Th31: :: EXTREAL1:31
canceled;
theorem Th32: :: EXTREAL1:32
for
x,
y being
R_eal st
y <> 0. holds
for
a,
b being
Real st
x = a &
y = b holds
x / y = a / b
theorem Th33: :: EXTREAL1:33
theorem Th34: :: EXTREAL1:34
:: deftheorem Def3 defines |. EXTREAL1:def 3 :
theorem Th35: :: EXTREAL1:35
canceled;
theorem Th36: :: EXTREAL1:36
theorem Th37: :: EXTREAL1:37
theorem Th38: :: EXTREAL1:38
theorem Th39: :: EXTREAL1:39
theorem Th40: :: EXTREAL1:40
theorem Th41: :: EXTREAL1:41
theorem Th42: :: EXTREAL1:42
theorem Th43: :: EXTREAL1:43
theorem Th44: :: EXTREAL1:44
theorem Th45: :: EXTREAL1:45
theorem Th46: :: EXTREAL1:46
for
x,
y being
R_eal st
x is
Real &
y is
Real holds
(
x < y iff ex
p,
q being
Real st
(
p = x &
q = y &
p < q ) ) ;
theorem Th47: :: EXTREAL1:47
theorem Th48: :: EXTREAL1:48
theorem Th49: :: EXTREAL1:49
theorem Th50: :: EXTREAL1:50
theorem Th51: :: EXTREAL1:51
theorem Th52: :: EXTREAL1:52