Article REAL_2, MML version 4.99.1005

:: REAL_2:th 63
theorem
for b1, b2 being real set holds
ex b3 being real set st
   b1 = b2 - b3;

:: REAL_2:th 105
theorem
for b1, b2, b3 being real set
      st (0 < b1 + - b2 & b2 - b1 < 0 & b2 + - b1 < 0 & b2 + - b3 < b1 - b3 & b2 - b3 < b1 + - b3 implies b3 - b2 <= b3 - b1)
   holds b1 <= b2;

:: REAL_2:th 106
theorem
for b1, b2, b3 being real set
      st (0 <= b1 + - b2 & b2 - b1 <= 0 & (- b1) + b2 <= 0 & b2 + - b3 <= b1 - b3 & b2 - b3 <= b1 + - b3 implies b3 - b2 < b3 - b1)
   holds b1 < b2;

:: REAL_2:th 109
theorem
for b1, b2 being real set
      st b1 <= - b2
   holds b1 + b2 <= 0 & b2 <= - b1;

:: REAL_2:th 110
theorem
for b1, b2 being real set
      st b1 < - b2
   holds b1 + b2 < 0 & b2 < - b1;

:: REAL_2:th 117
theorem
for b1, b2 being real set
      st 0 < b1
   holds (1 < b2 / b1 implies b1 < b2) &
    (b2 / b1 < 1 implies b2 < b1) &
    (b2 / b1 <= - 1 or - b1 < b2 & - b2 < b1) &
    (- 1 <= b2 / b1 or b2 < - b1 & b1 < - b2);

:: REAL_2:th 118
theorem
for b1, b2 being real set
      st 0 < b1
   holds (1 <= b2 / b1 implies b1 <= b2) &
    (b2 / b1 <= 1 implies b2 <= b1) &
    (- 1 <= b2 / b1 implies - b1 <= b2 & - b2 <= b1) &
    (b2 / b1 <= - 1 implies b2 <= - b1 & b1 <= - b2);

:: REAL_2:th 119
theorem
for b1, b2 being real set
      st b1 < 0
   holds (1 < b2 / b1 implies b2 < b1) &
    (b2 / b1 < 1 implies b1 < b2) &
    (b2 / b1 <= - 1 or b2 < - b1 & b1 < - b2) &
    (- 1 <= b2 / b1 or - b1 < b2 & - b2 < b1);

:: REAL_2:th 120
theorem
for b1, b2 being real set
      st b1 < 0
   holds (1 <= b2 / b1 implies b2 <= b1) &
    (b2 / b1 <= 1 implies b1 <= b2) &
    (- 1 <= b2 / b1 implies b2 <= - b1 & b1 <= - b2) &
    (b2 / b1 <= - 1 implies - b1 <= b2 & - b2 <= b1);

:: REAL_2:th 121
theorem
for b1, b2 being real set
      st (0 <= b1 & 0 <= b2 or b1 <= 0 & b2 <= 0)
   holds 0 <= b1 * b2;

:: REAL_2:th 122
theorem
for b1, b2 being real set
      st (b1 < 0 & b2 < 0 or 0 < b1 & 0 < b2)
   holds 0 < b1 * b2;

:: REAL_2:th 125
theorem
for b1, b2 being real set
      st (b1 <= 0 & b2 <= 0 or 0 <= b1 & 0 <= b2)
   holds 0 <= b1 / b2;

:: REAL_2:th 126
theorem
for b1, b2 being real set
      st (0 <= b1 & b2 < 0 or b1 <= 0 & 0 < b2)
   holds b1 / b2 <= 0;

:: REAL_2:th 127
theorem
for b1, b2 being real set
      st (0 < b1 & 0 < b2 or b1 < 0 & b2 < 0)
   holds 0 < b1 / b2;

:: REAL_2:th 128
theorem
for b1, b2 being real set
      st b1 < 0 & 0 < b2
   holds b1 / b2 < 0 & b2 / b1 < 0;

:: REAL_2:th 129
theorem
for b1, b2 being real set
      st b1 * b2 <= 0 & (0 <= b1 implies 0 < b2)
   holds b1 <= 0 & 0 <= b2;

:: REAL_2:th 133
theorem
for b1, b2 being real set
      st b1 <> 0 & b2 / b1 <= 0 & (0 < b1 implies 0 < b2)
   holds b1 < 0 & 0 <= b2;

:: REAL_2:th 134
theorem
for b1, b2 being real set
      st b1 <> 0 & 0 <= b2 / b1 & (0 < b1 implies b2 < 0)
   holds b1 < 0 & b2 <= 0;

:: REAL_2:th 137
theorem
for b1, b2 being real set
      st (1 < b1 & (b2 <= 1 implies 1 <= b2) or b1 < - 1 & (- 1 <= b2 implies b2 <= - 1))
   holds 1 < b1 * b2;

:: REAL_2:th 138
theorem
for b1, b2 being real set
      st (1 <= b1 & 1 <= b2 or b1 <= - 1 & b2 <= - 1)
   holds 1 <= b1 * b2;

:: REAL_2:th 139
theorem
for b1, b2 being real set
      st (0 <= b1 & b1 < 1 & 0 <= b2 & b2 <= 1 or b1 <= 0 & - 1 < b1 & b2 <= 0 & - 1 <= b2)
   holds b1 * b2 < 1;

:: REAL_2:th 140
theorem
for b1, b2 being real set
      st (0 <= b1 & b1 <= 1 & 0 <= b2 & b2 <= 1 or b1 <= 0 & - 1 <= b1 & b2 <= 0 & - 1 <= b2)
   holds b1 * b2 <= 1;

:: REAL_2:th 142
theorem
for b1, b2 being real set
      st (0 < b1 & b1 < b2 or b2 < b1 & b1 < 0)
   holds b1 / b2 < 1 & 1 < b2 / b1;

:: REAL_2:th 143
theorem
for b1, b2 being real set
      st (0 < b1 & b1 <= b2 or b2 <= b1 & b1 < 0)
   holds b1 / b2 <= 1 & 1 <= b2 / b1;

:: REAL_2:th 144
theorem
for b1, b2 being real set
      st (0 < b1 & 1 < b2 or b1 < 0 & b2 < 1)
   holds b1 < b1 * b2;

:: REAL_2:th 145
theorem
for b1, b2 being real set
      st (0 < b1 & b2 < 1 or b1 < 0 & 1 < b2)
   holds b1 * b2 < b1;

:: REAL_2:th 146
theorem
for b1, b2 being real set
      st (0 <= b1 & 1 <= b2 or b1 <= 0 & b2 <= 1)
   holds b1 <= b1 * b2;

:: REAL_2:th 147
theorem
for b1, b2 being real set
      st (0 <= b1 & b2 <= 1 or b1 <= 0 & 1 <= b2)
   holds b1 * b2 <= b1;

:: REAL_2:th 149
theorem
for b1 being real set
      st b1 < 0
   holds 1 / b1 < 0 & b1 " < 0;

:: REAL_2:th 150
theorem
for b1 being real set holds
   (1 / b1 < 0 implies b1 < 0) &
    (0 < 1 / b1 implies 0 < b1);

:: REAL_2:th 151
theorem
for b1, b2 being real set
      st (b1 <= 0 implies b2 < 0) & b1 < b2
   holds 1 / b2 < 1 / b1;

:: REAL_2:th 152
theorem
for b1, b2 being real set
      st (b1 <= 0 implies b2 < 0) & b1 <= b2
   holds 1 / b2 <= 1 / b1;

:: REAL_2:th 153
theorem
for b1, b2 being real set
      st b1 < 0 & 0 < b2
   holds 1 / b1 < 1 / b2;

:: REAL_2:th 154
theorem
for b1, b2 being real set
      st (1 / b1 <= 0 implies 1 / b2 < 0) &
         1 / b1 < 1 / b2
   holds b2 < b1;

:: REAL_2:th 155
theorem
for b1, b2 being real set
      st (1 / b1 <= 0 implies 1 / b2 < 0) &
         1 / b1 <= 1 / b2
   holds b2 <= b1;

:: REAL_2:th 156
theorem
for b1, b2 being real set
      st 1 / b1 < 0 & 0 < 1 / b2
   holds b1 < b2;

:: REAL_2:th 157
theorem
for b1 being real set
      st b1 < - 1
   holds - 1 < 1 / b1;

:: REAL_2:th 158
theorem
for b1 being real set
      st b1 <= - 1
   holds - 1 <= 1 / b1;

:: REAL_2:th 164
theorem
for b1 being real set
      st 1 <= b1
   holds 1 / b1 <= 1;

:: REAL_2:th 165
theorem
for b1, b2, b3 being real set holds
(b1 <= b2 - b3 implies b3 <= b2 - b1) & (b2 - b3 <= b1 implies b2 - b1 <= b3);

:: REAL_2:th 169
theorem
for b1, b2, b3, b4 being real set
      st b1 - b2 <= b3 - b4
   holds b1 + b4 <= b3 + b2 & b1 - b3 <= b2 - b4 & b4 - b2 <= b3 - b1 & b4 - b3 <= b2 - b1;

:: REAL_2:th 170
theorem
for b1, b2, b3, b4 being real set
      st b1 - b2 < b3 - b4
   holds b1 + b4 < b3 + b2 & b1 - b3 < b2 - b4 & b4 - b2 < b3 - b1 & b4 - b3 < b2 - b1;

:: REAL_2:th 171
theorem
for b1, b2, b3, b4 being real set holds
(b1 + b2 <= b3 - b4 implies b1 + b4 <= b3 - b2) &
 (b3 - b4 <= b1 + b2 implies b3 - b2 <= b1 + b4);

:: REAL_2:th 173
theorem
for b1, b2 being real set holds
(0 <= b1 or b1 + b2 < b2 & b2 < b2 - b1) &
 ((b2 <= b1 + b2 implies b2 < b2 - b1) implies b1 < 0);

:: REAL_2:th 174
theorem
for b1, b2 being real set holds
(b1 <= 0 implies b1 + b2 <= b2 & b2 <= b2 - b1) &
 (b2 < b1 + b2 & b2 - b1 < b2 or b1 <= 0);

:: REAL_2:th 177
theorem
for b1, b2, b3 being real set holds
(0 < b1 & b2 * b1 <= b3 implies b2 <= b3 / b1) &
 (b1 < 0 & b2 * b1 <= b3 implies b3 / b1 <= b2) &
 (0 < b1 & b3 <= b2 * b1 implies b3 / b1 <= b2) &
 (b1 < 0 & b3 <= b2 * b1 implies b2 <= b3 / b1);

:: REAL_2:th 178
theorem
for b1, b2, b3 being real set holds
(0 < b1 & b2 * b1 < b3 implies b2 < b3 / b1) &
 (b1 < 0 & b2 * b1 < b3 implies b3 / b1 < b2) &
 (0 < b1 & b3 < b2 * b1 implies b3 / b1 < b2) &
 (b1 < 0 & b3 < b2 * b1 implies b2 < b3 / b1);

:: REAL_2:th 181
theorem
for b1, b2 being real set
      st (for b3 being real set
              st 0 < b3
           holds b2 <= b1 + b3 or for b3 being real set
              st b3 < 0
           holds b2 <= b1 - b3)
   holds b2 <= b1;

:: REAL_2:th 182
theorem
for b1, b2 being real set
      st (for b3 being real set
              st 0 < b3
           holds b1 - b3 <= b2 or for b3 being real set
              st b3 < 0
           holds b1 + b3 <= b2)
   holds b1 <= b2;

:: REAL_2:th 183
theorem
for b1, b2 being real set
      st (for b3 being real set
              st 1 < b3
           holds b2 <= b1 * b3 or for b3 being real set
              st 0 < b3 & b3 < 1
           holds b2 <= b1 / b3)
   holds b2 <= b1;

:: REAL_2:th 184
theorem
for b1, b2 being real set
      st (for b3 being real set
              st 0 < b3 & b3 < 1
           holds b1 * b3 <= b2 or for b3 being real set
              st 1 < b3
           holds b1 / b3 <= b2)
   holds b1 <= b2;

:: REAL_2:th 185
theorem
for b1, b2, b3, b4 being real set
      st (0 < b1 & 0 < b2 or b1 < 0 & b2 < 0) &
         b3 * b2 < b4 * b1
   holds b3 / b1 < b4 / b2;

:: REAL_2:th 186
theorem
for b1, b2, b3, b4 being real set
      st (0 < b1 & b2 < 0 or b1 < 0 & 0 < b2) &
         b3 * b2 < b4 * b1
   holds b4 / b2 < b3 / b1;

:: REAL_2:th 187
theorem
for b1, b2, b3, b4 being real set
      st (0 < b1 & 0 < b2 or b1 < 0 & b2 < 0) &
         b3 * b2 <= b4 * b1
   holds b3 / b1 <= b4 / b2;

:: REAL_2:th 188
theorem
for b1, b2, b3, b4 being real set
      st (0 < b1 & b2 < 0 or b1 < 0 & 0 < b2) &
         b3 * b2 <= b4 * b1
   holds b4 / b2 <= b3 / b1;

:: REAL_2:th 193
theorem
for b1, b2, b3, b4 being real set
      st (b1 < 0 & b2 < 0 or 0 < b1 & 0 < b2)
   holds (b3 * b1 < b4 / b2 implies b3 * b2 < b4 / b1) &
    (b4 / b2 < b3 * b1 implies b4 / b1 < b3 * b2);

:: REAL_2:th 194
theorem
for b1, b2, b3, b4 being real set
      st (b1 < 0 & 0 < b2 or 0 < b1 & b2 < 0)
   holds (b3 * b1 < b4 / b2 implies b4 / b1 < b3 * b2) &
    (b4 / b2 < b3 * b1 implies b3 * b2 < b4 / b1);

:: REAL_2:th 199
theorem
for b1, b2, b3, b4 being real set
      st (0 < b1 & b1 <= b2 & 0 < b3 & b3 < b4 or b1 < 0 & b2 <= b1 & b3 < 0 & b4 < b3)
   holds b1 * b3 < b2 * b4;

:: REAL_2:th 200
theorem
for b1, b2, b3 being real set holds
(0 < b1 & 0 < b2 & b2 < b3 implies b1 / b3 < b1 / b2) &
 (0 < b1 & b3 < 0 & b2 < b3 implies b1 / b3 < b1 / b2);

:: REAL_2:th 201
theorem
for b1, b2, b3 being real set
      st 0 <= b1 & (b2 <= 0 implies b3 < 0) & b2 <= b3
   holds b1 / b3 <= b1 / b2;

:: REAL_2:th 202
theorem
for b1, b2, b3 being real set
      st b1 < 0 & (b2 <= 0 implies b3 < 0) & b2 < b3
   holds b1 / b2 < b1 / b3;

:: REAL_2:th 203
theorem
for b1, b2, b3 being real set
      st b1 <= 0 & (b2 <= 0 implies b3 < 0) & b2 <= b3
   holds b1 / b2 <= b1 / b3;

:: REAL_2:th 204
theorem
for b1, b2 being Element of bool REAL
      st b1 <> {} &
         b2 <> {} &
         (for b3, b4 being real set
               st b3 in b1 & b4 in b2
            holds b3 <= b4)
   holds ex b3 being real set st
      (for b4 being real set
             st b4 in b1
          holds b4 <= b3) &
       (for b4 being real set
             st b4 in b2
          holds b3 <= b4);