:: XREAL_1 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 )
theorem Th1: :: XREAL_1:1
theorem Th2: :: XREAL_1:2
Lemma5:
for b1 being real number
for b2, b3 being Element of REAL holds
( b1 = [*b2,b3*] implies ( b3 = 0 & b1 = b2 ) )
Lemma6:
for b1, b2 being Element of REAL
for b3, b4 being real number holds
( b1 = b3 & b2 = b4 implies + b1,b2 = b3 + b4 )
Lemma7:
{} in {{} }
by TARSKI:def 1;
Lemma8:
for b1, b2, b3 being real number holds
( b1 <= b2 implies b1 + b3 <= b2 + b3 )
Lemma9:
for b1, b2, b3, b4 being real number holds
( b1 <= b2 & b3 <= b4 implies b1 + b3 <= b2 + b4 )
Lemma10:
for b1, b2, b3 being real number holds
( b1 <= b2 implies b1 - b3 <= b2 - b3 )
Lemma11:
for b1, b2, b3, b4 being real number holds
not ( b1 < b2 & b3 <= b4 & not b1 + b3 < b2 + b4 )
Lemma12:
for b1, b2 being real number holds
not ( 0 < b1 & not b2 < b2 + b1 )
theorem Th3: :: XREAL_1:3
theorem Th4: :: XREAL_1:4
theorem Th5: :: XREAL_1:5
Lemma13:
for b1, b2, b3 being real number holds
( b1 + b2 <= b3 + b2 implies b1 <= b3 )
theorem Th6: :: XREAL_1:6
Lemma14:
for b1, b2 being Element of REAL
for b3, b4 being real number holds
( b1 = b3 & b2 = b4 implies * b1,b2 = b3 * b4 )
reconsider c1 = 0 as Element of REAL+ by ARYTM_2:21;
Lemma15:
for b1, b2, b3 being real number holds
( b1 <= b2 & 0 <= b3 implies b1 * b3 <= b2 * b3 )
Lemma16:
for b1, b2, b3 being real number holds
not ( 0 < b1 & b2 < b3 & not b2 * b1 < b3 * b1 )
theorem Th7: :: XREAL_1:7
theorem Th8: :: XREAL_1:8
theorem Th9: :: XREAL_1:9
theorem Th10: :: XREAL_1:10
Lemma18:
for b1, b2 being real number holds
( b1 <= b2 implies - b2 <= - b1 )
Lemma19:
for b1, b2 being real number holds
( - b1 <= - b2 implies b2 <= b1 )
theorem Th11: :: XREAL_1:11
theorem Th12: :: XREAL_1:12
Lemma21:
for b1, b2, b3 being real number holds
( b1 + b2 <= b3 implies b1 <= b3 - b2 )
Lemma22:
for b1, b2, b3 being real number holds
( b1 <= b2 - b3 implies b1 + b3 <= b2 )
Lemma23:
for b1, b2, b3 being real number holds
( b1 <= b2 + b3 implies b1 - b2 <= b3 )
Lemma24:
for b1, b2, b3 being real number holds
( b1 - b2 <= b3 implies b1 <= b2 + b3 )
theorem Th13: :: XREAL_1:13
theorem Th14: :: XREAL_1:14
theorem Th15: :: XREAL_1:15
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 <= b
2 & b
3 <= b
4 implies b
1 - b
4 <= b
2 - b
3 )
theorem Th16: :: XREAL_1:16
for b
1, b
2, b
3, b
4 being
real number holds
not ( b
1 < b
2 & b
3 <= b
4 & not b
1 - b
4 < b
2 - b
3 )
theorem Th17: :: XREAL_1:17
for b
1, b
2, b
3, b
4 being
real number holds
not ( b
1 <= b
2 & b
3 < b
4 & not b
1 - b
4 < b
2 - b
3 )
Lemma27:
for b1, b2, b3, b4 being real number holds
( b1 + b2 <= b3 + b4 implies b1 - b3 <= b4 - b2 )
theorem Th18: :: XREAL_1:18
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 - b
2 <= b
3 - b
4 implies b
1 - b
3 <= b
2 - b
4 )
theorem Th19: :: XREAL_1:19
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 - b
2 <= b
3 - b
4 implies b
4 - b
2 <= b
3 - b
1 )
theorem Th20: :: XREAL_1:20
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 - b
2 <= b
3 - b
4 implies b
4 - b
3 <= b
2 - b
1 )
theorem Th21: :: XREAL_1:21
theorem Th22: :: XREAL_1:22
theorem Th23: :: XREAL_1:23
theorem Th24: :: XREAL_1:24
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 + b
2 <= b
3 - b
4 implies b
1 + b
4 <= b
3 - b
2 )
theorem Th25: :: XREAL_1:25
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 - b
2 <= b
3 + b
4 implies b
1 - b
4 <= b
3 + b
2 )
theorem Th26: :: XREAL_1:26
Lemma28:
for b1, b2 being real number holds
not ( b1 < b2 & not 0 < b2 - b1 )
theorem Th27: :: XREAL_1:27
Lemma30:
for b1, b2 being real number holds
( b1 <= b2 implies 0 <= b2 - b1 )
theorem Th28: :: XREAL_1:28
theorem Th29: :: XREAL_1:29
for b
1, b
2 being
real number holds
( 0
<= b
1 & 0
<= b
2 & b
1 + b
2 = 0 implies b
1 = 0 )
theorem Th30: :: XREAL_1:30
for b
1, b
2 being
real number holds
( b
1 <= 0 & b
2 <= 0 & b
1 + b
2 = 0 implies b
1 = 0 )
theorem Th31: :: XREAL_1:31
theorem Th32: :: XREAL_1:32
for b
1, b
2 being
real number holds
not ( b
1 < 0 & not b
1 + b
2 < b
2 )
Lemma33:
for b1, b2 being real number holds
not ( b1 < b2 & not b1 - b2 < 0 )
Lemma34:
for b1 being real number holds
( not ( b1 < 0 & not 0 < - b1 ) & not ( 0 < - b1 & not b1 < 0 ) )
theorem Th33: :: XREAL_1:33
theorem Th34: :: XREAL_1:34
theorem Th35: :: XREAL_1:35
theorem Th36: :: XREAL_1:36
for b
1, b
2 being
real number holds
not ( 0
<= b
1 & 0
< b
2 & not 0
< b
1 + b
2 )
theorem Th37: :: XREAL_1:37
theorem Th38: :: XREAL_1:38
for b
1, b
2, b
3 being
real number holds
not ( b
1 <= 0 & b
2 < b
3 & not b
2 + b
1 < b
3 )
theorem Th39: :: XREAL_1:39
for b
1, b
2, b
3 being
real number holds
not ( b
1 < 0 & b
2 <= b
3 & not b
2 + b
1 < b
3 )
theorem Th40: :: XREAL_1:40
theorem Th41: :: XREAL_1:41
for b
1, b
2, b
3 being
real number holds
not ( 0
< b
1 & b
2 <= b
3 & not b
2 < b
1 + b
3 )
theorem Th42: :: XREAL_1:42
for b
1, b
2, b
3 being
real number holds
not ( 0
<= b
1 & b
2 < b
3 & not b
2 < b
1 + b
3 )
Lemma39:
for b1, b2, b3 being real number holds
not ( b1 < 0 & b2 < b3 & not b3 * b1 < b2 * b1 )
Lemma40:
for b1 being real number holds
not ( 0 < b1 & not 0 < b1 " )
Lemma41:
for b1, b2, b3 being real number holds
( 0 <= b1 & b2 <= b3 implies b2 / b1 <= b3 / b1 )
Lemma42:
for b1, b2, b3 being real number holds
not ( 0 < b1 & b2 < b3 & not b2 / b1 < b3 / b1 )
Lemma43:
for b1 being real number holds
not ( 0 < b1 & not 0 < b1 / 2 )
Lemma44:
for b1 being real number holds
not ( 0 < b1 & not b1 / 2 < b1 )
theorem Th43: :: XREAL_1:43
theorem Th44: :: XREAL_1:44
theorem Th45: :: XREAL_1:45
theorem Th46: :: XREAL_1:46
for b
1, b
2 being
real number holds
not ( 0
< b
1 & not b
2 - b
1 < b
2 )
theorem Th47: :: XREAL_1:47
theorem Th48: :: XREAL_1:48
for b
1, b
2 being
real number holds
not ( b
1 < 0 & not b
2 < b
2 - b
1 )
theorem Th49: :: XREAL_1:49
theorem Th50: :: XREAL_1:50
theorem Th51: :: XREAL_1:51
theorem Th52: :: XREAL_1:52
theorem Th53: :: XREAL_1:53
for b
1, b
2, b
3 being
real number holds
not ( 0
<= b
1 & b
2 < b
3 & not b
2 - b
1 < b
3 )
theorem Th54: :: XREAL_1:54
theorem Th55: :: XREAL_1:55
for b
1, b
2, b
3 being
real number holds
not ( b
1 <= 0 & b
2 < b
3 & not b
2 < b
3 - b
1 )
theorem Th56: :: XREAL_1:56
for b
1, b
2, b
3 being
real number holds
not ( b
1 < 0 & b
2 <= b
3 & not b
2 < b
3 - b
1 )
theorem Th57: :: XREAL_1:57
for b
1, b
2 being
real number holds
not ( b
1 <> b
2 & not 0
< b
1 - b
2 & not 0
< b
2 - b
1 )
theorem Th58: :: XREAL_1:58
theorem Th59: :: XREAL_1:59
theorem Th60: :: XREAL_1:60
theorem Th61: :: XREAL_1:61
theorem Th62: :: XREAL_1:62
theorem Th63: :: XREAL_1:63
for b
1, b
2 being
real number holds
not ( b
1 < - b
2 & not b
1 + b
2 < 0 )
theorem Th64: :: XREAL_1:64
for b
1, b
2 being
real number holds
not (
- b
1 < b
2 & not 0
< b
2 + b
1 )
Lemma46:
for b1, b2, b3 being real number holds
( b1 <= b2 & b3 <= 0 implies b2 * b3 <= b1 * b3 )
theorem Th65: :: XREAL_1:65
theorem Th66: :: XREAL_1:66
theorem Th67: :: XREAL_1:67
theorem Th68: :: XREAL_1:68
for b
1, b
2, b
3, b
4 being
real number holds
( 0
<= b
1 & b
1 <= b
2 & 0
<= b
3 & b
3 <= b
4 implies b
1 * b
3 <= b
2 * b
4 )
theorem Th69: :: XREAL_1:69
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 <= 0 & b
2 <= b
1 & b
3 <= 0 & b
4 <= b
3 implies b
1 * b
3 <= b
2 * b
4 )
theorem Th70: :: XREAL_1:70
theorem Th71: :: XREAL_1:71
theorem Th72: :: XREAL_1:72
for b
1, b
2, b
3, b
4 being
real number holds
not ( b
1 < 0 & b
2 <= b
1 & b
3 < 0 & b
4 < b
3 & not b
1 * b
3 < b
2 * b
4 )
theorem Th73: :: XREAL_1:73
for b
1, b
2, b
3, b
4 being
real number holds
not ( 0
<= b
1 & 0
<= b
2 & 0
<= b
3 & 0
<= b
4 &
(b1 * b3) + (b2 * b4) = 0 & not b
1 = 0 & not b
3 = 0 )
Lemma49:
for b1, b2, b3 being real number holds
not ( b1 < 0 & b2 < b3 & not b3 / b1 < b2 / b1 )
Lemma50:
for b1, b2, b3 being real number holds
not ( b1 <= 0 & b2 / b1 < b3 / b1 & not b3 < b2 )
theorem Th74: :: XREAL_1:74
theorem Th75: :: XREAL_1:75
theorem Th76: :: XREAL_1:76
theorem Th77: :: XREAL_1:77
theorem Th78: :: XREAL_1:78
for b
1, b
2, b
3 being
real number holds
not ( 0
< b
1 & 0
< b
2 & b
2 < b
3 & not b
1 / b
3 < b
1 / b
2 )
Lemma51:
for b1, b2 being real number holds
not ( b1 < 0 & 0 < b2 & not b2 / b1 < 0 )
Lemma52:
for b1, b2 being real number holds
not ( b1 < 0 & b2 < b1 & not b1 " < b2 " )
theorem Th79: :: XREAL_1:79
for b
1, b
2, b
3 being
real number holds
( 0
< b
1 & b
2 * b
1 <= b
3 implies b
2 <= b
3 / b
1 )
theorem Th80: :: XREAL_1:80
for b
1, b
2, b
3 being
real number holds
( b
1 < 0 & b
2 * b
1 <= b
3 implies b
3 / b
1 <= b
2 )
theorem Th81: :: XREAL_1:81
for b
1, b
2, b
3 being
real number holds
( 0
< b
1 & b
2 <= b
3 * b
1 implies b
2 / b
1 <= b
3 )
theorem Th82: :: XREAL_1:82
for b
1, b
2, b
3 being
real number holds
( b
1 < 0 & b
2 <= b
3 * b
1 implies b
3 <= b
2 / b
1 )
theorem Th83: :: XREAL_1:83
for b
1, b
2, b
3 being
real number holds
not ( 0
< b
1 & b
2 * b
1 < b
3 & not b
2 < b
3 / b
1 )
theorem Th84: :: XREAL_1:84
for b
1, b
2, b
3 being
real number holds
not ( b
1 < 0 & b
2 * b
1 < b
3 & not b
3 / b
1 < b
2 )
theorem Th85: :: XREAL_1:85
for b
1, b
2, b
3 being
real number holds
not ( 0
< b
1 & b
2 < b
3 * b
1 & not b
2 / b
1 < b
3 )
theorem Th86: :: XREAL_1:86
for b
1, b
2, b
3 being
real number holds
not ( b
1 < 0 & b
2 < b
3 * b
1 & not b
3 < b
2 / b
1 )
Lemma61:
for b1, b2 being real number holds
( 0 < b1 & b1 <= b2 implies b2 " <= b1 " )
Lemma62:
for b1, b2 being real number holds
( b1 < 0 & b2 <= b1 implies b1 " <= b2 " )
theorem Th87: :: XREAL_1:87
theorem Th88: :: XREAL_1:88
theorem Th89: :: XREAL_1:89
Lemma63:
for b1, b2 being real number holds
not ( 0 < b1 & 0 < b2 & not 0 < b1 / b2 )
Lemma64:
for b1, b2 being real number holds
not ( 0 < b1 & b1 < b2 & not b2 " < b1 " )
theorem Th90: :: XREAL_1:90
theorem Th91: :: XREAL_1:91
theorem Th92: :: XREAL_1:92
theorem Th93: :: XREAL_1:93
theorem Th94: :: XREAL_1:94
Lemma65:
for b1, b2 being positive real number holds
0 < b1 * b2
;
Lemma66:
for b1, b2 being negative real number holds
0 < b1 * b2
;
Lemma67:
for b1, b2 being non negative real number holds 0 <= b1 * b2
;
Lemma68:
for b1, b2 being non positive real number holds 0 <= b1 * b2
;
Lemma69:
for b1 being real number
for b2 being non positive non negative real number holds 0 = b2 * b1
;
Lemma70:
for b1 being positive real number
for b2 being negative real number holds
b1 * b2 < 0
;
theorem Th95: :: XREAL_1:95
theorem Th96: :: XREAL_1:96
for b
1, b
2 being
real number holds
( 0
<= b
1 &
(b2 - b1) * (b2 + b1) < 0 implies (
- b
1 < b
2 & b
2 < b
1 ) )
theorem Th97: :: XREAL_1:97
for b
1, b
2 being
real number holds
not ( 0
<= b
1 & 0
<= (b2 - b1) * (b2 + b1) & not b
2 <= - b
1 & not b
1 <= b
2 )
theorem Th98: :: XREAL_1:98
for b
1, b
2, b
3, b
4 being
real number holds
not ( 0
<= b
1 & 0
<= b
2 & b
1 < b
3 & b
2 < b
4 & not b
1 * b
2 < b
3 * b
4 )
theorem Th99: :: XREAL_1:99
for b
1, b
2, b
3, b
4 being
real number holds
not ( 0
<= b
1 & 0
< b
2 & b
1 < b
3 & b
2 <= b
4 & not b
1 * b
2 < b
3 * b
4 )
theorem Th100: :: XREAL_1:100
for b
1, b
2, b
3, b
4 being
real number holds
not ( 0
< b
1 & 0
< b
2 & b
1 <= b
3 & b
2 < b
4 & not b
1 * b
2 < b
3 * b
4 )
theorem Th101: :: XREAL_1:101
for b
1, b
2, b
3 being
real number holds
not ( 0
< b
1 & b
2 < 0 & b
3 < b
2 & not b
1 / b
2 < b
1 / b
3 )
theorem Th102: :: XREAL_1:102
for b
1, b
2, b
3 being
real number holds
not ( b
1 < 0 & 0
< b
2 & b
2 < b
3 & not b
1 / b
2 < b
1 / b
3 )
theorem Th103: :: XREAL_1:103
for b
1, b
2, b
3 being
real number holds
not ( b
1 < 0 & b
2 < 0 & b
3 < b
2 & not b
1 / b
3 < b
1 / b
2 )
theorem Th104: :: XREAL_1:104
for b
1, b
2, b
3, b
4 being
real number holds
( 0
< b
1 & 0
< b
2 & b
3 * b
2 <= b
4 * b
1 implies b
3 / b
1 <= b
4 / b
2 )
theorem Th105: :: XREAL_1:105
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 < 0 & b
2 < 0 & b
3 * b
2 <= b
4 * b
1 implies b
3 / b
1 <= b
4 / b
2 )
theorem Th106: :: XREAL_1:106
for b
1, b
2, b
3, b
4 being
real number holds
( 0
< b
1 & b
2 < 0 & b
3 * b
2 <= b
4 * b
1 implies b
4 / b
2 <= b
3 / b
1 )
theorem Th107: :: XREAL_1:107
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 < 0 & 0
< b
2 & b
3 * b
2 <= b
4 * b
1 implies b
4 / b
2 <= b
3 / b
1 )
theorem Th108: :: XREAL_1:108
for b
1, b
2, b
3, b
4 being
real number holds
not ( 0
< b
1 & 0
< b
2 & b
3 * b
2 < b
4 * b
1 & not b
3 / b
1 < b
4 / b
2 )
theorem Th109: :: XREAL_1:109
for b
1, b
2, b
3, b
4 being
real number holds
not ( b
1 < 0 & b
2 < 0 & b
3 * b
2 < b
4 * b
1 & not b
3 / b
1 < b
4 / b
2 )
theorem Th110: :: XREAL_1:110
for b
1, b
2, b
3, b
4 being
real number holds
not ( 0
< b
1 & b
2 < 0 & b
3 * b
2 < b
4 * b
1 & not b
4 / b
2 < b
3 / b
1 )
theorem Th111: :: XREAL_1:111
for b
1, b
2, b
3, b
4 being
real number holds
not ( b
1 < 0 & 0
< b
2 & b
3 * b
2 < b
4 * b
1 & not b
4 / b
2 < b
3 / b
1 )
theorem Th112: :: XREAL_1:112
for b
1, b
2, b
3, b
4 being
real number holds
not ( b
1 < 0 & b
2 < 0 & b
3 * b
1 < b
4 / b
2 & not b
3 * b
2 < b
4 / b
1 )
theorem Th113: :: XREAL_1:113
for b
1, b
2, b
3, b
4 being
real number holds
not ( 0
< b
1 & 0
< b
2 & b
3 * b
1 < b
4 / b
2 & not b
3 * b
2 < b
4 / b
1 )
theorem Th114: :: XREAL_1:114
for b
1, b
2, b
3, b
4 being
real number holds
not ( b
1 < 0 & b
2 < 0 & b
3 / b
2 < b
4 * b
1 & not b
3 / b
1 < b
4 * b
2 )
theorem Th115: :: XREAL_1:115
for b
1, b
2, b
3, b
4 being
real number holds
not ( 0
< b
1 & 0
< b
2 & b
3 / b
2 < b
4 * b
1 & not b
3 / b
1 < b
4 * b
2 )
theorem Th116: :: XREAL_1:116
for b
1, b
2, b
3, b
4 being
real number holds
not ( b
1 < 0 & 0
< b
2 & b
3 * b
1 < b
4 / b
2 & not b
4 / b
1 < b
3 * b
2 )
theorem Th117: :: XREAL_1:117
for b
1, b
2, b
3, b
4 being
real number holds
not ( 0
< b
1 & b
2 < 0 & b
3 * b
1 < b
4 / b
2 & not b
4 / b
1 < b
3 * b
2 )
theorem Th118: :: XREAL_1:118
for b
1, b
2, b
3, b
4 being
real number holds
not ( b
1 < 0 & 0
< b
2 & b
3 / b
2 < b
4 * b
1 & not b
4 * b
2 < b
3 / b
1 )
theorem Th119: :: XREAL_1:119
for b
1, b
2, b
3, b
4 being
real number holds
not ( 0
< b
1 & b
2 < 0 & b
3 / b
2 < b
4 * b
1 & not b
4 * b
2 < b
3 / b
1 )
theorem Th120: :: XREAL_1:120
for b
1, b
2, b
3 being
real number holds
( 0
< b
1 & 0
<= b
2 & b
1 <= b
3 implies b
2 / b
3 <= b
2 / b
1 )
theorem Th121: :: XREAL_1:121
for b
1, b
2, b
3 being
real number holds
( 0
<= b
1 & b
2 < 0 & b
3 <= b
2 implies b
1 / b
2 <= b
1 / b
3 )
theorem Th122: :: XREAL_1:122
for b
1, b
2, b
3 being
real number holds
( b
1 <= 0 & 0
< b
2 & b
2 <= b
3 implies b
1 / b
2 <= b
1 / b
3 )
theorem Th123: :: XREAL_1:123
for b
1, b
2, b
3 being
real number holds
( b
1 <= 0 & b
2 < 0 & b
3 <= b
2 implies b
1 / b
3 <= b
1 / b
2 )
theorem Th124: :: XREAL_1:124
for b
1 being
real number holds
( not ( 0
< b
1 & not 0
< b
1 " ) & not ( 0
< b
1 " & not 0
< b
1 ) )
theorem Th125: :: XREAL_1:125
for b
1 being
real number holds
( not ( b
1 < 0 & not b
1 " < 0 ) & not ( b
1 " < 0 & not b
1 < 0 ) )
theorem Th126: :: XREAL_1:126
for b
1, b
2 being
real number holds
not ( b
1 < 0 & 0
< b
2 & not b
1 " < b
2 " )
theorem Th127: :: XREAL_1:127
for b
1, b
2 being
real number holds
not ( b
1 " < 0 & 0
< b
2 " & not b
1 < b
2 )
theorem Th128: :: XREAL_1:128
theorem Th129: :: XREAL_1:129
theorem Th130: :: XREAL_1:130
theorem Th131: :: XREAL_1:131
theorem Th132: :: XREAL_1:132
theorem Th133: :: XREAL_1:133
theorem Th134: :: XREAL_1:134
theorem Th135: :: XREAL_1:135
for b
1, b
2 being
real number holds
not ( b
1 * b
2 < 0 & not ( b
1 > 0 & b
2 < 0 ) & not ( b
1 < 0 & b
2 > 0 ) )
theorem Th136: :: XREAL_1:136
for b
1, b
2 being
real number holds
not ( b
1 * b
2 > 0 & not ( b
1 > 0 & b
2 > 0 ) & not ( b
1 < 0 & b
2 < 0 ) )
theorem Th137: :: XREAL_1:137
theorem Th138: :: XREAL_1:138
theorem Th139: :: XREAL_1:139
theorem Th140: :: XREAL_1:140
theorem Th141: :: XREAL_1:141
theorem Th142: :: XREAL_1:142
for b
1, b
2 being
real number holds
not ( b
1 < 0 & b
2 < 0 & not 0
< b
1 / b
2 )
theorem Th143: :: XREAL_1:143
for b
1, b
2 being
real number holds
not ( b
1 < 0 & 0
< b
2 & not b
1 / b
2 < 0 )
theorem Th144: :: XREAL_1:144
theorem Th145: :: XREAL_1:145
for b
1, b
2 being
real number holds
not ( b
1 / b
2 < 0 & not ( b
2 < 0 & b
1 > 0 ) & not ( b
2 > 0 & b
1 < 0 ) )
theorem Th146: :: XREAL_1:146
for b
1, b
2 being
real number holds
not ( b
1 / b
2 > 0 & not ( b
2 > 0 & b
1 > 0 ) & not ( b
2 < 0 & b
1 < 0 ) )
theorem Th147: :: XREAL_1:147
theorem Th148: :: XREAL_1:148
theorem Th149: :: XREAL_1:149
theorem Th150: :: XREAL_1:150
theorem Th151: :: XREAL_1:151
theorem Th152: :: XREAL_1:152
for b
1, b
2 being
real number holds
( 0
<= b
1 & b
1 <= 1 & 0
<= b
2 & b
2 <= 1 & b
1 * b
2 = 1 implies b
1 = 1 )
theorem Th153: :: XREAL_1:153
theorem Th154: :: XREAL_1:154
theorem Th155: :: XREAL_1:155
theorem Th156: :: XREAL_1:156
theorem Th157: :: XREAL_1:157
for b
1, b
2 being
real number holds
not ( 0
< b
1 & 1
< b
2 & not b
1 < b
1 * b
2 )
theorem Th158: :: XREAL_1:158
for b
1, b
2 being
real number holds
not ( b
1 < 0 & b
2 < 1 & not b
1 < b
1 * b
2 )
theorem Th159: :: XREAL_1:159
for b
1, b
2 being
real number holds
not ( 0
< b
1 & b
2 < 1 & not b
1 * b
2 < b
1 )
theorem Th160: :: XREAL_1:160
for b
1, b
2 being
real number holds
not ( b
1 < 0 & 1
< b
2 & not b
1 * b
2 < b
1 )
theorem Th161: :: XREAL_1:161
theorem Th162: :: XREAL_1:162
theorem Th163: :: XREAL_1:163
for b
1, b
2 being
real number holds
not ( 1
< b
1 & 1
<= b
2 & not 1
< b
1 * b
2 )
theorem Th164: :: XREAL_1:164
for b
1, b
2 being
real number holds
not ( 0
<= b
1 & b
1 < 1 & 0
<= b
2 & b
2 <= 1 & not b
1 * b
2 < 1 )
theorem Th165: :: XREAL_1:165
theorem Th166: :: XREAL_1:166
theorem Th167: :: XREAL_1:167
theorem Th168: :: XREAL_1:168
theorem Th169: :: XREAL_1:169
Lemma80:
for b1 being real number holds
not ( 1 < b1 & not b1 " < 1 )
theorem Th170: :: XREAL_1:170
Lemma81:
for b1, b2 being real number holds
( b1 <= b2 & 0 <= b1 implies b1 / b2 <= 1 )
Lemma82:
for b1, b2 being real number holds
( b1 <= b2 & 0 <= b2 implies b1 / b2 <= 1 )
theorem Th171: :: XREAL_1:171
theorem Th172: :: XREAL_1:172
for b
1, b
2, b
3 being
real number holds
not ( 0
<= b
1 & b
1 <= 1 & 0
<= b
2 & 0
<= b
3 &
(b1 * b2) + ((1 - b1) * b3) = 0 & not ( b
1 = 0 & b
3 = 0 ) & not ( b
1 = 1 & b
2 = 0 ) & not ( b
2 = 0 & b
3 = 0 ) )
theorem Th173: :: XREAL_1:173
for b
1, b
2, b
3 being
real number holds
( 0
<= b
1 & b
1 <= 1 & b
2 <= b
3 implies b
2 <= ((1 - b1) * b2) + (b1 * b3) )
theorem Th174: :: XREAL_1:174
for b
1, b
2, b
3 being
real number holds
( 0
<= b
1 & b
1 <= 1 & b
2 <= b
3 implies
((1 - b1) * b2) + (b1 * b3) <= b
3 )
theorem Th175: :: XREAL_1:175
for b
1, b
2, b
3, b
4 being
real number holds
( 0
<= b
1 & b
1 <= 1 & b
2 <= b
3 & b
2 <= b
4 implies b
2 <= ((1 - b1) * b3) + (b1 * b4) )
theorem Th176: :: XREAL_1:176
for b
1, b
2, b
3, b
4 being
real number holds
( 0
<= b
1 & b
1 <= 1 & b
2 <= b
3 & b
4 <= b
3 implies
((1 - b1) * b2) + (b1 * b4) <= b
3 )
theorem Th177: :: XREAL_1:177
for b
1, b
2, b
3, b
4 being
real number holds
not ( 0
<= b
1 & b
1 <= 1 & b
2 < b
3 & b
2 < b
4 & not b
2 < ((1 - b1) * b3) + (b1 * b4) )
theorem Th178: :: XREAL_1:178
for b
1, b
2, b
3, b
4 being
real number holds
not ( 0
<= b
1 & b
1 <= 1 & b
2 < b
3 & b
4 < b
3 & not
((1 - b1) * b2) + (b1 * b4) < b
3 )
theorem Th179: :: XREAL_1:179
for b
1, b
2, b
3, b
4 being
real number holds
not ( 0
< b
1 & b
1 < 1 & b
2 <= b
3 & b
2 < b
4 & not b
2 < ((1 - b1) * b3) + (b1 * b4) )
theorem Th180: :: XREAL_1:180
for b
1, b
2, b
3, b
4 being
real number holds
not ( 0
< b
1 & b
1 < 1 & b
2 < b
3 & b
4 <= b
3 & not
((1 - b1) * b2) + (b1 * b4) < b
3 )
theorem Th181: :: XREAL_1:181
for b
1, b
2, b
3 being
real number holds
( 0
<= b
1 & b
1 <= 1 & b
2 <= ((1 - b1) * b2) + (b1 * b3) & b
3 < ((1 - b1) * b2) + (b1 * b3) implies b
1 = 0 )
theorem Th182: :: XREAL_1:182
for b
1, b
2, b
3 being
real number holds
( 0
<= b
1 & b
1 <= 1 &
((1 - b1) * b2) + (b1 * b3) <= b
2 &
((1 - b1) * b2) + (b1 * b3) < b
3 implies b
1 = 0 )
theorem Th183: :: XREAL_1:183
theorem Th184: :: XREAL_1:184
theorem Th185: :: XREAL_1:185
theorem Th186: :: XREAL_1:186
theorem Th187: :: XREAL_1:187
theorem Th188: :: XREAL_1:188
theorem Th189: :: XREAL_1:189
for b
1, b
2 being
real number holds
not ( 0
< b
1 & b
1 < b
2 & not 1
< b
2 / b
1 )
theorem Th190: :: XREAL_1:190
for b
1, b
2 being
real number holds
not ( b
1 < 0 & b
2 < b
1 & not 1
< b
2 / b
1 )
theorem Th191: :: XREAL_1:191
for b
1, b
2 being
real number holds
not ( 0
<= b
1 & b
1 < b
2 & not b
1 / b
2 < 1 )
theorem Th192: :: XREAL_1:192
for b
1, b
2 being
real number holds
not ( b
1 <= 0 & b
2 < b
1 & not b
1 / b
2 < 1 )
theorem Th193: :: XREAL_1:193
for b
1, b
2 being
real number holds
not ( 0
< b
1 & b
2 < b
1 & not b
2 / b
1 < 1 )
theorem Th194: :: XREAL_1:194
for b
1, b
2 being
real number holds
not ( b
1 < 0 & b
1 < b
2 & not b
2 / b
1 < 1 )
theorem Th195: :: XREAL_1:195
theorem Th196: :: XREAL_1:196
theorem Th197: :: XREAL_1:197
theorem Th198: :: XREAL_1:198
theorem Th199: :: XREAL_1:199
for b
1, b
2 being
real number holds
not ( 0
< b
1 &
- b
1 < b
2 & not
- 1
< b
2 / b
1 )
theorem Th200: :: XREAL_1:200
for b
1, b
2 being
real number holds
not ( 0
< b
1 &
- b
2 < b
1 & not
- 1
< b
2 / b
1 )
theorem Th201: :: XREAL_1:201
for b
1, b
2 being
real number holds
not ( b
1 < 0 & b
2 < - b
1 & not
- 1
< b
2 / b
1 )
theorem Th202: :: XREAL_1:202
for b
1, b
2 being
real number holds
not ( b
1 < 0 & b
1 < - b
2 & not
- 1
< b
2 / b
1 )
theorem Th203: :: XREAL_1:203
theorem Th204: :: XREAL_1:204
theorem Th205: :: XREAL_1:205
theorem Th206: :: XREAL_1:206
theorem Th207: :: XREAL_1:207
for b
1, b
2 being
real number holds
not ( 0
< b
1 & b
2 < - b
1 & not b
2 / b
1 < - 1 )
theorem Th208: :: XREAL_1:208
for b
1, b
2 being
real number holds
not ( 0
< b
1 & b
1 < - b
2 & not b
2 / b
1 < - 1 )
theorem Th209: :: XREAL_1:209
for b
1, b
2 being
real number holds
not ( b
1 < 0 &
- b
1 < b
2 & not b
2 / b
1 < - 1 )
theorem Th210: :: XREAL_1:210
for b
1, b
2 being
real number holds
not ( b
1 < 0 &
- b
2 < b
1 & not b
2 / b
1 < - 1 )
theorem Th211: :: XREAL_1:211
theorem Th212: :: XREAL_1:212
theorem Th213: :: XREAL_1:213
theorem Th214: :: XREAL_1:214
theorem Th215: :: XREAL_1:215
theorem Th216: :: XREAL_1:216
theorem Th217: :: XREAL_1:217
theorem Th218: :: XREAL_1:218
theorem Th219: :: XREAL_1:219
theorem Th220: :: XREAL_1:220
theorem Th221: :: XREAL_1:221
theorem Th222: :: XREAL_1:222
theorem Th223: :: XREAL_1:223
theorem Th224: :: XREAL_1:224
theorem Th225: :: XREAL_1:225
theorem Th226: :: XREAL_1:226
theorem Th227: :: XREAL_1:227