Article XREAL_1, MML version 4.99.1005

:: XREAL_1:th 3
theorem
for b1 being real set holds
   ex b2 being real set st
      b1 < b2;

:: XREAL_1:th 4
theorem
for b1 being real set holds
   ex b2 being real set st
      b2 < b1;

:: XREAL_1:th 5
theorem
for b1, b2 being real set holds
ex b3 being real set st
   b1 < b3 & b2 < b3;

:: XREAL_1:th 6
theorem
for b1, b2 being real set holds
ex b3 being real set st
   b3 < b1 & b3 < b2;

:: XREAL_1:th 7
theorem
for b1, b2 being real set
      st b1 < b2
   holds ex b3 being real set st
      b1 < b3 & b3 < b2;

:: XREAL_1:th 8
theorem
for b1, b2, b3 being real set holds
   b1 <= b2
iff
   b1 + b3 <= b2 + b3;

:: XREAL_1:th 9
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b3 <= b4
   holds b1 + b3 <= b2 + b4;

:: XREAL_1:th 10
theorem
for b1, b2, b3, b4 being real set
      st b1 < b2 & b3 <= b4
   holds b1 + b3 < b2 + b4;

:: XREAL_1:th 11
theorem
for b1, b2, b3 being real set holds
   b1 <= b2
iff
   b1 - b3 <= b2 - b3;

:: XREAL_1:th 12
theorem
for b1, b2, b3 being real set holds
   b1 <= b2
iff
   b3 - b2 <= b3 - b1;

:: XREAL_1:th 13
theorem
for b1, b2, b3 being real set
      st b1 <= b2 - b3
   holds b3 <= b2 - b1;

:: XREAL_1:th 14
theorem
for b1, b2, b3 being real set
      st b1 - b2 <= b3
   holds b1 - b3 <= b2;

:: XREAL_1:th 15
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b3 <= b4
   holds b1 - b4 <= b2 - b3;

:: XREAL_1:th 16
theorem
for b1, b2, b3, b4 being real set
      st b1 < b2 & b3 <= b4
   holds b1 - b4 < b2 - b3;

:: XREAL_1:th 17
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b3 < b4
   holds b1 - b4 < b2 - b3;

:: XREAL_1:th 18
theorem
for b1, b2, b3, b4 being real set
      st b1 - b2 <= b3 - b4
   holds b1 - b3 <= b2 - b4;

:: XREAL_1:th 19
theorem
for b1, b2, b3, b4 being real set
      st b1 - b2 <= b3 - b4
   holds b4 - b2 <= b3 - b1;

:: XREAL_1:th 20
theorem
for b1, b2, b3, b4 being real set
      st b1 - b2 <= b3 - b4
   holds b4 - b3 <= b2 - b1;

:: XREAL_1:th 21
theorem
for b1, b2, b3 being real set holds
   b1 + b2 <= b3
iff
   b1 <= b3 - b2;

:: XREAL_1:th 22
theorem
for b1, b2, b3 being real set holds
   b1 <= b2 + b3
iff
   b1 - b2 <= b3;

:: XREAL_1:th 23
theorem
for b1, b2, b3, b4 being real set holds
   b1 + b2 <= b3 + b4
iff
   b1 - b3 <= b4 - b2;

:: XREAL_1:th 24
theorem
for b1, b2, b3, b4 being real set
      st b1 + b2 <= b3 - b4
   holds b1 + b4 <= b3 - b2;

:: XREAL_1:th 25
theorem
for b1, b2, b3, b4 being real set
      st b1 - b2 <= b3 + b4
   holds b1 - b4 <= b3 + b2;

:: XREAL_1:th 26
theorem
for b1, b2 being real set holds
   b1 <= b2
iff
   - b2 <= - b1;

:: XREAL_1:th 27
theorem
for b1, b2 being real set
      st b1 <= - b2
   holds b2 <= - b1;

:: XREAL_1:th 28
theorem
for b1, b2 being real set
      st - b1 <= b2
   holds - b2 <= b1;

:: XREAL_1:th 29
theorem
for b1, b2 being real set
      st 0 <= b1 & 0 <= b2 & b1 + b2 = 0
   holds b1 = 0;

:: XREAL_1:th 30
theorem
for b1, b2 being real set
      st b1 <= 0 & b2 <= 0 & b1 + b2 = 0
   holds b1 = 0;

:: XREAL_1:th 31
theorem
for b1, b2 being real set
      st 0 < b1
   holds b2 < b2 + b1;

:: XREAL_1:th 32
theorem
for b1, b2 being real set
      st b1 < 0
   holds b1 + b2 < b2;

:: XREAL_1:th 33
theorem
for b1, b2 being real set
      st 0 <= b1
   holds b2 <= b1 + b2;

:: XREAL_1:th 34
theorem
for b1, b2 being real set
      st b1 <= 0
   holds b1 + b2 <= b2;

:: XREAL_1:th 35
theorem
for b1, b2 being real set
      st 0 <= b1 & 0 <= b2
   holds 0 <= b1 + b2;

:: XREAL_1:th 36
theorem
for b1, b2 being real set
      st 0 <= b1 & 0 < b2
   holds 0 < b1 + b2;

:: XREAL_1:th 37
theorem
for b1, b2, b3 being real set
      st b1 <= 0 & b2 <= b3
   holds b2 + b1 <= b3;

:: XREAL_1:th 38
theorem
for b1, b2, b3 being real set
      st b1 <= 0 & b2 < b3
   holds b2 + b1 < b3;

:: XREAL_1:th 39
theorem
for b1, b2, b3 being real set
      st b1 < 0 & b2 <= b3
   holds b2 + b1 < b3;

:: XREAL_1:th 40
theorem
for b1, b2, b3 being real set
      st 0 <= b1 & b2 <= b3
   holds b2 <= b1 + b3;

:: XREAL_1:th 41
theorem
for b1, b2, b3 being real set
      st 0 < b1 & b2 <= b3
   holds b2 < b1 + b3;

:: XREAL_1:th 42
theorem
for b1, b2, b3 being real set
      st 0 <= b1 & b2 < b3
   holds b2 < b1 + b3;

:: XREAL_1:th 43
theorem
for b1, b2 being real set
      st for b3 being real set
              st 0 < b3
           holds b1 <= b2 + b3
   holds b1 <= b2;

:: XREAL_1:th 44
theorem
for b1, b2 being real set
      st for b3 being real set
              st b3 < 0
           holds b1 + b3 <= b2
   holds b1 <= b2;

:: XREAL_1:th 45
theorem
for b1, b2 being real set
      st 0 <= b1
   holds b2 - b1 <= b2;

:: XREAL_1:th 46
theorem
for b1, b2 being real set
      st 0 < b1
   holds b2 - b1 < b2;

:: XREAL_1:th 47
theorem
for b1, b2 being real set
      st b1 <= 0
   holds b2 <= b2 - b1;

:: XREAL_1:th 48
theorem
for b1, b2 being real set
      st b1 < 0
   holds b2 < b2 - b1;

:: XREAL_1:th 49
theorem
for b1, b2 being real set
      st b1 <= b2
   holds b1 - b2 <= 0;

:: XREAL_1:th 50
theorem
for b1, b2 being real set
      st b1 <= b2
   holds 0 <= b2 - b1;

:: XREAL_1:th 51
theorem
for b1, b2 being real set
      st b1 < b2
   holds b1 - b2 < 0;

:: XREAL_1:th 52
theorem
for b1, b2 being real set
      st b1 < b2
   holds 0 < b2 - b1;

:: XREAL_1:th 53
theorem
for b1, b2, b3 being real set
      st 0 <= b1 & b2 < b3
   holds b2 - b1 < b3;

:: XREAL_1:th 54
theorem
for b1, b2, b3 being real set
      st b1 <= 0 & b2 <= b3
   holds b2 <= b3 - b1;

:: XREAL_1:th 55
theorem
for b1, b2, b3 being real set
      st b1 <= 0 & b2 < b3
   holds b2 < b3 - b1;

:: XREAL_1:th 56
theorem
for b1, b2, b3 being real set
      st b1 < 0 & b2 <= b3
   holds b2 < b3 - b1;

:: XREAL_1:th 57
theorem
for b1, b2 being real set
      st b1 <> b2 & b1 - b2 <= 0
   holds 0 < b2 - b1;

:: XREAL_1:th 58
theorem
for b1, b2 being real set
      st for b3 being real set
              st b3 < 0
           holds b1 <= b2 - b3
   holds b1 <= b2;

:: XREAL_1:th 59
theorem
for b1, b2 being real set
      st for b3 being real set
              st 0 < b3
           holds b1 - b3 <= b2
   holds b1 <= b2;

:: XREAL_1:th 60
theorem
for b1 being real set holds
      b1 < 0
   iff
      0 < - b1;

:: XREAL_1:th 61
theorem
for b1, b2 being real set
      st b1 <= - b2
   holds b1 + b2 <= 0;

:: XREAL_1:th 62
theorem
for b1, b2 being real set
      st - b1 <= b2
   holds 0 <= b1 + b2;

:: XREAL_1:th 63
theorem
for b1, b2 being real set
      st b1 < - b2
   holds b1 + b2 < 0;

:: XREAL_1:th 64
theorem
for b1, b2 being real set
      st - b1 < b2
   holds 0 < b2 + b1;

:: XREAL_1:th 65
theorem
for b1 being real set holds
   0 <= b1 * b1;

:: XREAL_1:th 66
theorem
for b1, b2, b3 being real set
      st b1 <= b2 & 0 <= b3
   holds b1 * b3 <= b2 * b3;

:: XREAL_1:th 67
theorem
for b1, b2, b3 being real set
      st b1 <= b2 & b3 <= 0
   holds b2 * b3 <= b1 * b3;

:: XREAL_1:th 68
theorem
for b1, b2, b3, b4 being real set
      st 0 <= b1 & b1 <= b2 & 0 <= b3 & b3 <= b4
   holds b1 * b3 <= b2 * b4;

:: XREAL_1:th 69
theorem
for b1, b2, b3, b4 being real set
      st b1 <= 0 & b2 <= b1 & b3 <= 0 & b4 <= b3
   holds b1 * b3 <= b2 * b4;

:: XREAL_1:th 70
theorem
for b1, b2, b3 being real set
      st 0 < b1 & b2 < b3
   holds b2 * b1 < b3 * b1;

:: XREAL_1:th 71
theorem
for b1, b2, b3 being real set
      st b1 < 0 & b2 < b3
   holds b3 * b1 < b2 * b1;

:: XREAL_1:th 72
theorem
for b1, b2, b3, b4 being real set
      st b1 < 0 & b2 <= b1 & b3 < 0 & b4 < b3
   holds b1 * b3 < b2 * b4;

:: XREAL_1:th 73
theorem
for b1, b2, b3, b4 being real set
      st 0 <= b1 & 0 <= b2 & 0 <= b3 & 0 <= b4 & (b1 * b3) + (b2 * b4) = 0 & b1 <> 0
   holds b3 = 0;

:: XREAL_1:th 74
theorem
for b1, b2, b3 being real set
      st 0 <= b1 & b2 <= b3
   holds b2 / b1 <= b3 / b1;

:: XREAL_1:th 75
theorem
for b1, b2, b3 being real set
      st b1 <= 0 & b2 <= b3
   holds b3 / b1 <= b2 / b1;

:: XREAL_1:th 76
theorem
for b1, b2, b3 being real set
      st 0 < b1 & b2 < b3
   holds b2 / b1 < b3 / b1;

:: XREAL_1:th 77
theorem
for b1, b2, b3 being real set
      st b1 < 0 & b2 < b3
   holds b3 / b1 < b2 / b1;

:: XREAL_1:th 78
theorem
for b1, b2, b3 being real set
      st 0 < b1 & 0 < b2 & b2 < b3
   holds b1 / b3 < b1 / b2;

:: XREAL_1:th 79
theorem
for b1, b2, b3 being real set
      st 0 < b1 & b2 * b1 <= b3
   holds b2 <= b3 / b1;

:: XREAL_1:th 80
theorem
for b1, b2, b3 being real set
      st b1 < 0 & b2 * b1 <= b3
   holds b3 / b1 <= b2;

:: XREAL_1:th 81
theorem
for b1, b2, b3 being real set
      st 0 < b1 & b2 <= b3 * b1
   holds b2 / b1 <= b3;

:: XREAL_1:th 82
theorem
for b1, b2, b3 being real set
      st b1 < 0 & b2 <= b3 * b1
   holds b3 <= b2 / b1;

:: XREAL_1:th 83
theorem
for b1, b2, b3 being real set
      st 0 < b1 & b2 * b1 < b3
   holds b2 < b3 / b1;

:: XREAL_1:th 84
theorem
for b1, b2, b3 being real set
      st b1 < 0 & b2 * b1 < b3
   holds b3 / b1 < b2;

:: XREAL_1:th 85
theorem
for b1, b2, b3 being real set
      st 0 < b1 & b2 < b3 * b1
   holds b2 / b1 < b3;

:: XREAL_1:th 86
theorem
for b1, b2, b3 being real set
      st b1 < 0 & b2 < b3 * b1
   holds b3 < b2 / b1;

:: XREAL_1:th 87
theorem
for b1, b2 being real set
      st 0 < b1 & b1 <= b2
   holds b2 " <= b1 ";

:: XREAL_1:th 88
theorem
for b1, b2 being real set
      st b1 < 0 & b2 <= b1
   holds b1 " <= b2 ";

:: XREAL_1:th 89
theorem
for b1, b2 being real set
      st b1 < 0 & b2 < b1
   holds b1 " < b2 ";

:: XREAL_1:th 90
theorem
for b1, b2 being real set
      st 0 < b1 & b1 < b2
   holds b2 " < b1 ";

:: XREAL_1:th 91
theorem
for b1, b2 being real set
      st 0 < b1 " & b1 " <= b2 "
   holds b2 <= b1;

:: XREAL_1:th 92
theorem
for b1, b2 being real set
      st b1 " < 0 & b2 " <= b1 "
   holds b1 <= b2;

:: XREAL_1:th 93
theorem
for b1, b2 being real set
      st 0 < b1 " & b1 " < b2 "
   holds b2 < b1;

:: XREAL_1:th 94
theorem
for b1, b2 being real set
      st b1 " < 0 & b2 " < b1 "
   holds b1 < b2;

:: XREAL_1:th 95
theorem
for b1, b2 being real set
      st 0 <= b1 & (b2 - b1) * (b2 + b1) <= 0
   holds - b1 <= b2 & b2 <= b1;

:: XREAL_1:th 96
theorem
for b1, b2 being real set
      st 0 <= b1 &
         (b2 - b1) * (b2 + b1) < 0
   holds - b1 < b2 & b2 < b1;

:: XREAL_1:th 97
theorem
for b1, b2 being real set
      st 0 <= b1 & 0 <= (b2 - b1) * (b2 + b1) & - b1 < b2
   holds b1 <= b2;

:: XREAL_1:th 98
theorem
for b1, b2, b3, b4 being real set
      st 0 <= b1 & 0 <= b2 & b1 < b3 & b2 < b4
   holds b1 * b2 < b3 * b4;

:: XREAL_1:th 99
theorem
for b1, b2, b3, b4 being real set
      st 0 <= b1 & 0 < b2 & b1 < b3 & b2 <= b4
   holds b1 * b2 < b3 * b4;

:: XREAL_1:th 100
theorem
for b1, b2, b3, b4 being real set
      st 0 < b1 & 0 < b2 & b1 <= b3 & b2 < b4
   holds b1 * b2 < b3 * b4;

:: XREAL_1:th 101
theorem
for b1, b2, b3 being real set
      st 0 < b1 & b2 < 0 & b3 < b2
   holds b1 / b2 < b1 / b3;

:: XREAL_1:th 102
theorem
for b1, b2, b3 being real set
      st b1 < 0 & 0 < b2 & b2 < b3
   holds b1 / b2 < b1 / b3;

:: XREAL_1:th 103
theorem
for b1, b2, b3 being real set
      st b1 < 0 & b2 < 0 & b3 < b2
   holds b1 / b3 < b1 / b2;

:: XREAL_1:th 104
theorem
for b1, b2, b3, b4 being real set
      st 0 < b1 & 0 < b2 & b3 * b2 <= b4 * b1
   holds b3 / b1 <= b4 / b2;

:: XREAL_1:th 105
theorem
for b1, b2, b3, b4 being real set
      st b1 < 0 & b2 < 0 & b3 * b2 <= b4 * b1
   holds b3 / b1 <= b4 / b2;

:: XREAL_1:th 106
theorem
for b1, b2, b3, b4 being real set
      st 0 < b1 & b2 < 0 & b3 * b2 <= b4 * b1
   holds b4 / b2 <= b3 / b1;

:: XREAL_1:th 107
theorem
for b1, b2, b3, b4 being real set
      st b1 < 0 & 0 < b2 & b3 * b2 <= b4 * b1
   holds b4 / b2 <= b3 / b1;

:: XREAL_1:th 108
theorem
for b1, b2, b3, b4 being real set
      st 0 < b1 & 0 < b2 & b3 * b2 < b4 * b1
   holds b3 / b1 < b4 / b2;

:: XREAL_1:th 109
theorem
for b1, b2, b3, b4 being real set
      st b1 < 0 & b2 < 0 & b3 * b2 < b4 * b1
   holds b3 / b1 < b4 / b2;

:: XREAL_1:th 110
theorem
for b1, b2, b3, b4 being real set
      st 0 < b1 & b2 < 0 & b3 * b2 < b4 * b1
   holds b4 / b2 < b3 / b1;

:: XREAL_1:th 111
theorem
for b1, b2, b3, b4 being real set
      st b1 < 0 & 0 < b2 & b3 * b2 < b4 * b1
   holds b4 / b2 < b3 / b1;

:: XREAL_1:th 112
theorem
for b1, b2, b3, b4 being real set
      st b1 < 0 & b2 < 0 & b3 * b1 < b4 / b2
   holds b3 * b2 < b4 / b1;

:: XREAL_1:th 113
theorem
for b1, b2, b3, b4 being real set
      st 0 < b1 & 0 < b2 & b3 * b1 < b4 / b2
   holds b3 * b2 < b4 / b1;

:: XREAL_1:th 114
theorem
for b1, b2, b3, b4 being real set
      st b1 < 0 & b2 < 0 & b3 / b2 < b4 * b1
   holds b3 / b1 < b4 * b2;

:: XREAL_1:th 115
theorem
for b1, b2, b3, b4 being real set
      st 0 < b1 & 0 < b2 & b3 / b2 < b4 * b1
   holds b3 / b1 < b4 * b2;

:: XREAL_1:th 116
theorem
for b1, b2, b3, b4 being real set
      st b1 < 0 & 0 < b2 & b3 * b1 < b4 / b2
   holds b4 / b1 < b3 * b2;

:: XREAL_1:th 117
theorem
for b1, b2, b3, b4 being real set
      st 0 < b1 & b2 < 0 & b3 * b1 < b4 / b2
   holds b4 / b1 < b3 * b2;

:: XREAL_1:th 118
theorem
for b1, b2, b3, b4 being real set
      st b1 < 0 & 0 < b2 & b3 / b2 < b4 * b1
   holds b4 * b2 < b3 / b1;

:: XREAL_1:th 119
theorem
for b1, b2, b3, b4 being real set
      st 0 < b1 & b2 < 0 & b3 / b2 < b4 * b1
   holds b4 * b2 < b3 / b1;

:: XREAL_1:th 120
theorem
for b1, b2, b3 being real set
      st 0 < b1 & 0 <= b2 & b1 <= b3
   holds b2 / b3 <= b2 / b1;

:: XREAL_1:th 121
theorem
for b1, b2, b3 being real set
      st 0 <= b1 & b2 < 0 & b3 <= b2
   holds b1 / b2 <= b1 / b3;

:: XREAL_1:th 122
theorem
for b1, b2, b3 being real set
      st b1 <= 0 & 0 < b2 & b2 <= b3
   holds b1 / b2 <= b1 / b3;

:: XREAL_1:th 123
theorem
for b1, b2, b3 being real set
      st b1 <= 0 & b2 < 0 & b3 <= b2
   holds b1 / b3 <= b1 / b2;

:: XREAL_1:th 124
theorem
for b1 being real set holds
      0 < b1
   iff
      0 < b1 ";

:: XREAL_1:th 125
theorem
for b1 being real set holds
      b1 < 0
   iff
      b1 " < 0;

:: XREAL_1:th 126
theorem
for b1, b2 being real set
      st b1 < 0 & 0 < b2
   holds b1 " < b2 ";

:: XREAL_1:th 127
theorem
for b1, b2 being real set
      st b1 " < 0 & 0 < b2 "
   holds b1 < b2;

:: XREAL_1:th 128
theorem
for b1, b2 being real set
      st b1 <= 0 & 0 <= b1
   holds 0 = b1 * b2;

:: XREAL_1:th 129
theorem
for b1, b2 being real set
      st 0 <= b1 & 0 <= b2
   holds 0 <= b1 * b2;

:: XREAL_1:th 130
theorem
for b1, b2 being real set
      st b1 <= 0 & b2 <= 0
   holds 0 <= b1 * b2;

:: XREAL_1:th 131
theorem
for b1, b2 being real set
      st 0 < b1 & 0 < b2
   holds 0 < b1 * b2;

:: XREAL_1:th 132
theorem
for b1, b2 being real set
      st b1 < 0 & b2 < 0
   holds 0 < b1 * b2;

:: XREAL_1:th 133
theorem
for b1, b2 being real set
      st 0 <= b1 & b2 <= 0
   holds b1 * b2 <= 0;

:: XREAL_1:th 134
theorem
for b1, b2 being real set
      st 0 < b1 & b2 < 0
   holds b1 * b2 < 0;

:: XREAL_1:th 135
theorem
for b1, b2 being real set
      st b1 * b2 < 0 & (b1 <= 0 or 0 <= b2)
   holds b1 < 0 & 0 < b2;

:: XREAL_1:th 136
theorem
for b1, b2 being real set
      st 0 < b1 * b2 & (b1 <= 0 or b2 <= 0)
   holds b1 < 0 & b2 < 0;

:: XREAL_1:th 137
theorem
for b1, b2 being real set
      st b1 <= 0 & b2 <= 0
   holds 0 <= b1 / b2;

:: XREAL_1:th 138
theorem
for b1, b2 being real set
      st 0 <= b1 & 0 <= b2
   holds 0 <= b1 / b2;

:: XREAL_1:th 139
theorem
for b1, b2 being real set
      st 0 <= b1 & b2 <= 0
   holds b1 / b2 <= 0;

:: XREAL_1:th 140
theorem
for b1, b2 being real set
      st b1 <= 0 & 0 <= b2
   holds b1 / b2 <= 0;

:: XREAL_1:th 141
theorem
for b1, b2 being real set
      st 0 < b1 & 0 < b2
   holds 0 < b1 / b2;

:: XREAL_1:th 142
theorem
for b1, b2 being real set
      st b1 < 0 & b2 < 0
   holds 0 < b1 / b2;

:: XREAL_1:th 143
theorem
for b1, b2 being real set
      st b1 < 0 & 0 < b2
   holds b1 / b2 < 0;

:: XREAL_1:th 144
theorem
for b1, b2 being real set
      st b1 < 0 & 0 < b2
   holds b2 / b1 < 0;

:: XREAL_1:th 145
theorem
for b1, b2 being real set
      st b1 / b2 < 0 & (0 <= b2 or b1 <= 0)
   holds 0 < b2 & b1 < 0;

:: XREAL_1:th 146
theorem
for b1, b2 being real set
      st 0 < b1 / b2 & (b2 <= 0 or b1 <= 0)
   holds b2 < 0 & b1 < 0;

:: XREAL_1:th 147
theorem
for b1, b2 being real set
      st b1 <= b2
   holds b1 < b2 + 1;

:: XREAL_1:th 148
theorem
for b1 being real set holds
   b1 - 1 < b1;

:: XREAL_1:th 149
theorem
for b1, b2 being real set
      st b1 <= b2
   holds b1 - 1 < b2;

:: XREAL_1:th 150
theorem
for b1 being real set
      st - 1 < b1
   holds 0 < 1 + b1;

:: XREAL_1:th 151
theorem
for b1 being real set
      st b1 < 1
   holds 0 < 1 - b1;

:: XREAL_1:th 152
theorem
for b1, b2 being real set
      st 0 <= b1 & b1 <= 1 & 0 <= b2 & b2 <= 1 & b1 * b2 = 1
   holds b1 = 1;

:: XREAL_1:th 153
theorem
for b1, b2 being real set
      st 0 <= b1 & 1 <= b2
   holds b1 <= b1 * b2;

:: XREAL_1:th 154
theorem
for b1, b2 being real set
      st b1 <= 0 & b2 <= 1
   holds b1 <= b1 * b2;

:: XREAL_1:th 155
theorem
for b1, b2 being real set
      st 0 <= b1 & b2 <= 1
   holds b1 * b2 <= b1;

:: XREAL_1:th 156
theorem
for b1, b2 being real set
      st b1 <= 0 & 1 <= b2
   holds b1 * b2 <= b1;

:: XREAL_1:th 157
theorem
for b1, b2 being real set
      st 0 < b1 & 1 < b2
   holds b1 < b1 * b2;

:: XREAL_1:th 158
theorem
for b1, b2 being real set
      st b1 < 0 & b2 < 1
   holds b1 < b1 * b2;

:: XREAL_1:th 159
theorem
for b1, b2 being real set
      st 0 < b1 & b2 < 1
   holds b1 * b2 < b1;

:: XREAL_1:th 160
theorem
for b1, b2 being real set
      st b1 < 0 & 1 < b2
   holds b1 * b2 < b1;

:: XREAL_1:th 161
theorem
for b1, b2 being real set
      st 1 <= b1 & 1 <= b2
   holds 1 <= b1 * b2;

:: XREAL_1:th 162
theorem
for b1, b2 being real set
      st 0 <= b1 & b1 <= 1 & 0 <= b2 & b2 <= 1
   holds b1 * b2 <= 1;

:: XREAL_1:th 163
theorem
for b1, b2 being real set
      st 1 < b1 & 1 <= b2
   holds 1 < b1 * b2;

:: XREAL_1:th 164
theorem
for b1, b2 being real set
      st 0 <= b1 & b1 < 1 & 0 <= b2 & b2 <= 1
   holds b1 * b2 < 1;

:: XREAL_1:th 165
theorem
for b1, b2 being real set
      st b1 <= - 1 & b2 <= - 1
   holds 1 <= b1 * b2;

:: XREAL_1:th 166
theorem
for b1, b2 being real set
      st b1 < - 1 & b2 <= - 1
   holds 1 < b1 * b2;

:: XREAL_1:th 167
theorem
for b1, b2 being real set
      st b1 <= 0 & - 1 <= b1 & b2 <= 0 & - 1 <= b2
   holds b1 * b2 <= 1;

:: XREAL_1:th 168
theorem
for b1, b2 being real set
      st b1 <= 0 & - 1 < b1 & b2 <= 0 & - 1 <= b2
   holds b1 * b2 < 1;

:: XREAL_1:th 169
theorem
for b1, b2 being real set
      st for b3 being real set
              st 1 < b3
           holds b1 <= b2 * b3
   holds b1 <= b2;

:: XREAL_1:th 170
theorem
for b1, b2 being real set
      st for b3 being real set
              st 0 < b3 & b3 < 1
           holds b1 * b3 <= b2
   holds b1 <= b2;

:: XREAL_1:th 171
theorem
for b1, b2 being real set
      st for b3 being real set
              st 0 < b3 & b3 < 1
           holds b1 <= b3 * b2
   holds b1 <= 0;

:: XREAL_1:th 172
theorem
for b1, b2, b3 being real set
      st 0 <= b1 &
         b1 <= 1 &
         0 <= b2 &
         0 <= b3 &
         (b1 * b2) + ((1 - b1) * b3) = 0 &
         (b1 = 0 implies b3 <> 0) &
         (b1 = 1 implies b2 <> 0)
   holds b2 = 0 & b3 = 0;

:: XREAL_1:th 173
theorem
for b1, b2, b3 being real set
      st 0 <= b1 & b1 <= 1 & b2 <= b3
   holds b2 <= ((1 - b1) * b2) + (b1 * b3);

:: XREAL_1:th 174
theorem
for b1, b2, b3 being real set
      st 0 <= b1 & b1 <= 1 & b2 <= b3
   holds ((1 - b1) * b2) + (b1 * b3) <= b3;

:: XREAL_1:th 175
theorem
for b1, b2, b3, b4 being real set
      st 0 <= b1 & b1 <= 1 & b2 <= b3 & b2 <= b4
   holds b2 <= ((1 - b1) * b3) + (b1 * b4);

:: XREAL_1:th 176
theorem
for b1, b2, b3, b4 being real set
      st 0 <= b1 & b1 <= 1 & b2 <= b3 & b4 <= b3
   holds ((1 - b1) * b2) + (b1 * b4) <= b3;

:: XREAL_1:th 177
theorem
for b1, b2, b3, b4 being real set
      st 0 <= b1 & b1 <= 1 & b2 < b3 & b2 < b4
   holds b2 < ((1 - b1) * b3) + (b1 * b4);

:: XREAL_1:th 178
theorem
for b1, b2, b3, b4 being real set
      st 0 <= b1 & b1 <= 1 & b2 < b3 & b4 < b3
   holds ((1 - b1) * b2) + (b1 * b4) < b3;

:: XREAL_1:th 179
theorem
for b1, b2, b3, b4 being real set
      st 0 < b1 & b1 < 1 & b2 <= b3 & b2 < b4
   holds b2 < ((1 - b1) * b3) + (b1 * b4);

:: XREAL_1:th 180
theorem
for b1, b2, b3, b4 being real set
      st 0 < b1 & b1 < 1 & b2 < b3 & b4 <= b3
   holds ((1 - b1) * b2) + (b1 * b4) < b3;

:: XREAL_1:th 181
theorem
for b1, b2, b3 being real set
      st 0 <= b1 &
         b1 <= 1 &
         b2 <= ((1 - b1) * b2) + (b1 * b3) &
         b3 < ((1 - b1) * b2) + (b1 * b3)
   holds b1 = 0;

:: XREAL_1:th 182
theorem
for b1, b2, b3 being real set
      st 0 <= b1 &
         b1 <= 1 &
         ((1 - b1) * b2) + (b1 * b3) <= b2 &
         ((1 - b1) * b2) + (b1 * b3) < b3
   holds b1 = 0;

:: XREAL_1:th 183
theorem
for b1, b2 being real set
      st 0 < b1 & b1 <= b2
   holds 1 <= b2 / b1;

:: XREAL_1:th 184
theorem
for b1, b2 being real set
      st b1 < 0 & b2 <= b1
   holds 1 <= b2 / b1;

:: XREAL_1:th 185
theorem
for b1, b2 being real set
      st 0 <= b1 & b1 <= b2
   holds b1 / b2 <= 1;

:: XREAL_1:th 186
theorem
for b1, b2 being real set
      st b1 <= b2 & b2 <= 0
   holds b2 / b1 <= 1;

:: XREAL_1:th 187
theorem
for b1, b2 being real set
      st 0 <= b1 & b2 <= b1
   holds b2 / b1 <= 1;

:: XREAL_1:th 188
theorem
for b1, b2 being real set
      st b1 <= 0 & b1 <= b2
   holds b2 / b1 <= 1;

:: XREAL_1:th 189
theorem
for b1, b2 being real set
      st 0 < b1 & b1 < b2
   holds 1 < b2 / b1;

:: XREAL_1:th 190
theorem
for b1, b2 being real set
      st b1 < 0 & b2 < b1
   holds 1 < b2 / b1;

:: XREAL_1:th 191
theorem
for b1, b2 being real set
      st 0 <= b1 & b1 < b2
   holds b1 / b2 < 1;

:: XREAL_1:th 192
theorem
for b1, b2 being real set
      st b1 <= 0 & b2 < b1
   holds b1 / b2 < 1;

:: XREAL_1:th 193
theorem
for b1, b2 being real set
      st 0 < b1 & b2 < b1
   holds b2 / b1 < 1;

:: XREAL_1:th 194
theorem
for b1, b2 being real set
      st b1 < 0 & b1 < b2
   holds b2 / b1 < 1;

:: XREAL_1:th 195
theorem
for b1, b2 being real set
      st 0 <= b1 & - b1 <= b2
   holds - 1 <= b2 / b1;

:: XREAL_1:th 196
theorem
for b1, b2 being real set
      st 0 <= b1 & - b2 <= b1
   holds - 1 <= b2 / b1;

:: XREAL_1:th 197
theorem
for b1, b2 being real set
      st b1 <= 0 & b2 <= - b1
   holds - 1 <= b2 / b1;

:: XREAL_1:th 198
theorem
for b1, b2 being real set
      st b1 <= 0 & b1 <= - b2
   holds - 1 <= b2 / b1;

:: XREAL_1:th 199
theorem
for b1, b2 being real set
      st 0 < b1 & - b1 < b2
   holds - 1 < b2 / b1;

:: XREAL_1:th 200
theorem
for b1, b2 being real set
      st 0 < b1 & - b2 < b1
   holds - 1 < b2 / b1;

:: XREAL_1:th 201
theorem
for b1, b2 being real set
      st b1 < 0 & b2 < - b1
   holds - 1 < b2 / b1;

:: XREAL_1:th 202
theorem
for b1, b2 being real set
      st b1 < 0 & b1 < - b2
   holds - 1 < b2 / b1;

:: XREAL_1:th 203
theorem
for b1, b2 being real set
      st 0 < b1 & b2 <= - b1
   holds b2 / b1 <= - 1;

:: XREAL_1:th 204
theorem
for b1, b2 being real set
      st 0 < b1 & b1 <= - b2
   holds b2 / b1 <= - 1;

:: XREAL_1:th 205
theorem
for b1, b2 being real set
      st b1 < 0 & - b1 <= b2
   holds b2 / b1 <= - 1;

:: XREAL_1:th 206
theorem
for b1, b2 being real set
      st b1 < 0 & - b2 <= b1
   holds b2 / b1 <= - 1;

:: XREAL_1:th 207
theorem
for b1, b2 being real set
      st 0 < b1 & b2 < - b1
   holds b2 / b1 < - 1;

:: XREAL_1:th 208
theorem
for b1, b2 being real set
      st 0 < b1 & b1 < - b2
   holds b2 / b1 < - 1;

:: XREAL_1:th 209
theorem
for b1, b2 being real set
      st b1 < 0 & - b1 < b2
   holds b2 / b1 < - 1;

:: XREAL_1:th 210
theorem
for b1, b2 being real set
      st b1 < 0 & - b2 < b1
   holds b2 / b1 < - 1;

:: XREAL_1:th 211
theorem
for b1, b2 being real set
      st for b3 being real set
              st 0 < b3 & b3 < 1
           holds b1 <= b2 / b3
   holds b1 <= b2;

:: XREAL_1:th 212
theorem
for b1, b2 being real set
      st for b3 being real set
              st 1 < b3
           holds b1 / b3 <= b2
   holds b1 <= b2;

:: XREAL_1:th 213
theorem
for b1 being real set
      st 1 <= b1
   holds b1 " <= 1;

:: XREAL_1:th 214
theorem
for b1 being real set
      st 1 < b1
   holds b1 " < 1;

:: XREAL_1:th 215
theorem
for b1 being real set
      st b1 <= - 1
   holds - 1 <= b1 ";

:: XREAL_1:th 216
theorem
for b1 being real set
      st b1 < - 1
   holds - 1 < b1 ";

:: XREAL_1:th 217
theorem
for b1 being real set
      st 0 < b1
   holds 0 < b1 / 2;

:: XREAL_1:th 218
theorem
for b1 being real set
      st 0 < b1
   holds b1 / 2 < b1;

:: XREAL_1:th 219
theorem
for b1 being real set
      st b1 <= 1 / 2
   holds (2 * b1) - 1 <= 0;

:: XREAL_1:th 220
theorem
for b1 being real set
      st b1 <= 1 / 2
   holds 0 <= 1 - (2 * b1);

:: XREAL_1:th 221
theorem
for b1 being real set
      st 1 / 2 <= b1
   holds 0 <= (2 * b1) - 1;

:: XREAL_1:th 222
theorem
for b1 being real set
      st 1 / 2 <= b1
   holds 1 - (2 * b1) <= 0;

:: XREAL_1:th 223
theorem
for b1 being real set
      st 0 < b1
   holds b1 / 3 < b1;

:: XREAL_1:th 224
theorem
for b1 being real set
      st 0 < b1
   holds 0 < b1 / 3;

:: XREAL_1:th 225
theorem
for b1 being real set
      st 0 < b1
   holds b1 / 4 < b1;

:: XREAL_1:th 226
theorem
for b1 being real set
      st 0 < b1
   holds 0 < b1 / 4;

:: XREAL_1:th 227
theorem
for b1, b2 being real set
      st b2 <> 0
   holds ex b3 being real set st
      b1 = b2 / b3;

:: XREAL_1:th 228
theorem
for b1, b2 being real set
      st b1 < b2
   holds b1 < (b1 + b2) / 2 & (b1 + b2) / 2 < b2;

:: XREAL_1:condreg 1
registration
  cluster -> ext-real (Element of REAL);
end;

:: XREAL_1:th 229
theorem
for b1, b2 being ext-real set
      st b1 < b2
   holds ex b3 being ext-real set st
      b1 < b3 & b3 < b2;

:: XREAL_1:th 230
theorem
for b1, b2, b3 being ext-real set
      st b1 < b2 &
         (for b4 being ext-real set
               st b1 < b4 & b4 < b2
            holds b3 <= b4)
   holds b3 <= b1;

:: XREAL_1:th 231
theorem
for b1, b2, b3 being ext-real set
      st b1 < b2 &
         (for b4 being ext-real set
               st b1 < b4 & b4 < b2
            holds b4 <= b3)
   holds b2 <= b3;