:: XREAL_1 semantic presentation
Lemma23:
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;
Lemma26:
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 Th1: :: XREAL_1:1
theorem Th2: :: XREAL_1:2
Lemma30:
for a being real number
for x1, x2 being Element of REAL st a = [*x1,x2*] holds
( x2 = 0 & a = x1 )
Lemma34:
for a', b' being Element of REAL
for a, b being real number st a' = a & b' = b holds
+ a',b' = a + b
Lemma42:
{} in {{} }
by TARSKI:def 1;
Lemma43:
for a, b, c being real number st a <= b holds
a + c <= b + c
Lemma137:
for a, b, c, d being real number st a <= b & c <= d holds
a + c <= b + d
Lemma138:
for a, b, c being real number st a <= b holds
a - c <= b - c
Lemma139:
for a, b, c, d being real number st a < b & c <= d holds
a + c < b + d
Lemma140:
for a, b being real number st 0 < a holds
b < b + a
theorem Th3: :: XREAL_1:3
theorem Th4: :: XREAL_1:4
theorem Th5: :: XREAL_1:5
Lemma142:
for a, c, b being real number st a + c <= b + c holds
a <= b
theorem Th6: :: XREAL_1:6
Lemma143:
for a', b' being Element of REAL
for a, b being real number st a' = a & b' = b holds
* a',b' = a * b
reconsider o = 0 as Element of REAL+ by ARYTM_2:21;
Lemma145:
for a, b, c being real number st a <= b & 0 <= c holds
a * c <= b * c
Lemma146:
for c, a, b being real number st 0 < c & a < b holds
a * c < b * c
theorem Th7: :: XREAL_1:7
theorem Th8: :: XREAL_1:8
theorem Th9: :: XREAL_1:9
theorem Th10: :: XREAL_1:10
Lemma148:
for a, b being real number st a <= b holds
- b <= - a
Lemma149:
for b, a being real number st - b <= - a holds
a <= b
theorem Th11: :: XREAL_1:11
theorem Th12: :: XREAL_1:12
Lemma151:
for a, b, c being real number st a + b <= c holds
a <= c - b
Lemma152:
for a, b, c being real number st a <= b - c holds
a + c <= b
Lemma153:
for a, b, c being real number st a <= b + c holds
a - b <= c
Lemma154:
for a, b, c being real number st a - b <= c holds
a <= b + c
theorem Th13: :: XREAL_1:13
theorem Th14: :: XREAL_1:14
theorem Th15: :: XREAL_1:15
theorem Th16: :: XREAL_1:16
theorem Th17: :: XREAL_1:17
Lemma157:
for a, b, c, d being real number st a + b <= c + d holds
a - c <= d - b
theorem Th18: :: XREAL_1:18
theorem Th19: :: XREAL_1:19
theorem Th20: :: XREAL_1:20
theorem Th21: :: XREAL_1:21
theorem Th22: :: XREAL_1:22
theorem Th23: :: XREAL_1:23
theorem Th24: :: XREAL_1:24
theorem Th25: :: XREAL_1:25
theorem Th26: :: XREAL_1:26
Lemma158:
for a, b being real number st a < b holds
0 < b - a
theorem Th27: :: XREAL_1:27
Lemma160:
for a, b being real number st a <= b holds
0 <= b - a
theorem Th28: :: XREAL_1:28
theorem Th29: :: XREAL_1:29
theorem Th30: :: XREAL_1:30
theorem Th31: :: XREAL_1:31
theorem Th32: :: XREAL_1:32
Lemma163:
for a, b being real number st a < b holds
a - b < 0
Lemma164:
for a being real number holds
( a < 0 iff 0 < - a )
theorem Th33: :: XREAL_1:33
theorem Th34: :: XREAL_1:34
theorem Th35: :: XREAL_1:35
theorem Th36: :: XREAL_1:36
theorem Th37: :: XREAL_1:37
theorem Th38: :: XREAL_1:38
theorem Th39: :: XREAL_1:39
theorem Th40: :: XREAL_1:40
theorem Th41: :: XREAL_1:41
theorem Th42: :: XREAL_1:42
Lemma169:
for c, a, b being real number st c < 0 & a < b holds
b * c < a * c
Lemma170:
for a being real number st 0 < a holds
0 < a "
Lemma171:
for c, b, a being real number st 0 <= c & b <= a holds
b / c <= a / c
Lemma172:
for c, a, b being real number st 0 < c & a < b holds
a / c < b / c
Lemma173:
for a being real number st 0 < a holds
0 < a / 2
Lemma174:
for a being real number st 0 < a holds
a / 2 < a
theorem Th43: :: XREAL_1:43
theorem Th44: :: XREAL_1:44
theorem Th45: :: XREAL_1:45
theorem Th46: :: XREAL_1:46
theorem Th47: :: XREAL_1:47
theorem Th48: :: XREAL_1:48
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
theorem Th54: :: XREAL_1:54
theorem Th55: :: XREAL_1:55
theorem Th56: :: XREAL_1:56
theorem Th57: :: XREAL_1:57
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
theorem Th64: :: XREAL_1:64
Lemma176:
for a, b, c being real number st a <= b & c <= 0 holds
b * c <= a * c
theorem Th65: :: XREAL_1:65
theorem Th66: :: XREAL_1:66
theorem Th67: :: XREAL_1:67
theorem Th68: :: XREAL_1:68
theorem Th69: :: XREAL_1:69
theorem Th70: :: XREAL_1:70
theorem Th71: :: XREAL_1:71
theorem Th72: :: XREAL_1:72
theorem Th73: :: XREAL_1:73
Lemma179:
for c, a, b being real number st c < 0 & a < b holds
b / c < a / c
Lemma180:
for c, b, a being real number st c <= 0 & b / c < a / c holds
a < b
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
Lemma181:
for a, b being real number st a < 0 & 0 < b holds
b / a < 0
Lemma182:
for b, a being real number st b < 0 & a < b holds
b " < a "
theorem Th79: :: XREAL_1:79
theorem Th80: :: XREAL_1:80
theorem Th81: :: XREAL_1:81
theorem Th82: :: XREAL_1:82
theorem Th83: :: XREAL_1:83
theorem Th84: :: XREAL_1:84
theorem Th85: :: XREAL_1:85
theorem Th86: :: XREAL_1:86
Lemma191:
for a, b being real number st 0 < a & a <= b holds
b " <= a "
Lemma192:
for b, a being real number st b < 0 & a <= b holds
b " <= a "
theorem Th87: :: XREAL_1:87
theorem Th88: :: XREAL_1:88
theorem Th89: :: XREAL_1:89
Lemma193:
for a, b being real number st 0 < a & 0 < b holds
0 < a / b
Lemma194:
for a, b being real number st 0 < a & a < b holds
b " < a "
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
Lemma195:
for a, b being positive real number holds 0 < a * b
;
Lemma196:
for a, b being negative real number holds 0 < a * b
;
Lemma197:
for a, b being non negative real number holds 0 <= a * b
;
Lemma198:
for a, b being non positive real number holds 0 <= a * b
;
Lemma199:
for b being real number
for a being non positive non negative real number holds 0 = a * b
;
Lemma200:
for a being positive real number
for b being negative real number holds a * b < 0
;
theorem Th95: :: XREAL_1:95
theorem Th96: :: XREAL_1:96
theorem Th97: :: XREAL_1:97
theorem Th98: :: XREAL_1:98
theorem Th99: :: XREAL_1:99
theorem Th100: :: XREAL_1:100
theorem Th101: :: XREAL_1:101
theorem Th102: :: XREAL_1:102
theorem Th103: :: XREAL_1:103
theorem Th104: :: XREAL_1:104
theorem Th105: :: XREAL_1:105
theorem Th106: :: XREAL_1:106
theorem Th107: :: XREAL_1:107
theorem Th108: :: XREAL_1:108
theorem Th109: :: XREAL_1:109
theorem Th110: :: XREAL_1:110
theorem Th111: :: XREAL_1:111
theorem Th112: :: XREAL_1:112
theorem Th113: :: XREAL_1:113
theorem Th114: :: XREAL_1:114
theorem Th115: :: XREAL_1:115
theorem Th116: :: XREAL_1:116
theorem Th117: :: XREAL_1:117
theorem Th118: :: XREAL_1:118
theorem Th119: :: XREAL_1:119
theorem Th120: :: XREAL_1:120
theorem Th121: :: XREAL_1:121
theorem Th122: :: XREAL_1:122
theorem Th123: :: XREAL_1:123
theorem Th124: :: XREAL_1:124
theorem Th125: :: XREAL_1:125
theorem Th126: :: XREAL_1:126
theorem Th127: :: XREAL_1:127
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
a,
b being
real number holds
( not
a * b < 0 or (
a > 0 &
b < 0 ) or (
a < 0 &
b > 0 ) )
theorem Th136: :: XREAL_1:136
for
a,
b being
real number holds
( not
a * b > 0 or (
a > 0 &
b > 0 ) or (
a < 0 &
b < 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
theorem Th143: :: XREAL_1:143
theorem Th144: :: XREAL_1:144
theorem Th145: :: XREAL_1:145
for
a,
b being
real number holds
( not
a / b < 0 or (
b < 0 &
a > 0 ) or (
b > 0 &
a < 0 ) )
theorem Th146: :: XREAL_1:146
for
a,
b being
real number holds
( not
a / b > 0 or (
b > 0 &
a > 0 ) or (
b < 0 &
a < 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
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
theorem Th158: :: XREAL_1:158
theorem Th159: :: XREAL_1:159
theorem Th160: :: XREAL_1:160
theorem Th161: :: XREAL_1:161
theorem Th162: :: XREAL_1:162
theorem Th163: :: XREAL_1:163
theorem Th164: :: XREAL_1:164
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
Lemma210:
for a being real number st 1 < a holds
a " < 1
theorem Th170: :: XREAL_1:170
Lemma211:
for a, b being real number st a <= b & 0 <= a holds
a / b <= 1
Lemma212:
for b, a being real number st b <= a & 0 <= a holds
b / a <= 1
theorem Th171: :: XREAL_1:171
theorem Th172: :: XREAL_1:172
for
d,
a,
b being
real number st 0
<= d &
d <= 1 & 0
<= a & 0
<= b &
(d * a) + ((1 - d) * b) = 0 & not (
d = 0 &
b = 0 ) & not (
d = 1 &
a = 0 ) holds
(
a = 0 &
b = 0 )
theorem Th173: :: XREAL_1:173
theorem Th174: :: XREAL_1:174
theorem Th175: :: XREAL_1:175
theorem Th176: :: XREAL_1:176
theorem Th177: :: XREAL_1:177
theorem Th178: :: XREAL_1:178
theorem Th179: :: XREAL_1:179
for
d,
a,
b,
c being
real number st 0
< d &
d < 1 &
a <= b &
a < c holds
a < ((1 - d) * b) + (d * c)
theorem Th180: :: XREAL_1:180
for
d,
b,
a,
c being
real number st 0
< d &
d < 1 &
b < a &
c <= a holds
((1 - d) * b) + (d * c) < a
theorem Th181: :: XREAL_1:181
theorem Th182: :: XREAL_1:182
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
theorem Th190: :: XREAL_1:190
theorem Th191: :: XREAL_1:191
theorem Th192: :: XREAL_1:192
theorem Th193: :: XREAL_1:193
theorem Th194: :: XREAL_1:194
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
theorem Th200: :: XREAL_1:200
theorem Th201: :: XREAL_1:201
theorem Th202: :: XREAL_1:202
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
theorem Th208: :: XREAL_1:208
theorem Th209: :: XREAL_1:209
theorem Th210: :: XREAL_1:210
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
theorem Th228: :: XREAL_1:228