:: 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 )
proof end;

theorem Th1: :: XREAL_1:1
for b1, b2 being real number holds
( b1 <= b2 & b2 <= b1 implies b1 = b2 ) by XXREAL_0:1;

theorem Th2: :: XREAL_1:2
for b1, b2, b3 being real number holds
( b1 <= b2 & b2 <= b3 implies b1 <= b3 ) by XXREAL_0:2;

Lemma5: for b1 being real number
for b2, b3 being Element of REAL holds
( b1 = [*b2,b3*] implies ( b3 = 0 & b1 = b2 ) )
proof end;

Lemma6: for b1, b2 being Element of REAL
for b3, b4 being real number holds
( b1 = b3 & b2 = b4 implies + b1,b2 = b3 + b4 )
proof end;

Lemma7: {} in {{} }
by TARSKI:def 1;

Lemma8: for b1, b2, b3 being real number holds
( b1 <= b2 implies b1 + b3 <= b2 + b3 )
proof end;

Lemma9: for b1, b2, b3, b4 being real number holds
( b1 <= b2 & b3 <= b4 implies b1 + b3 <= b2 + b4 )
proof end;

Lemma10: for b1, b2, b3 being real number holds
( b1 <= b2 implies b1 - b3 <= b2 - b3 )
proof end;

Lemma11: for b1, b2, b3, b4 being real number holds
not ( b1 < b2 & b3 <= b4 & not b1 + b3 < b2 + b4 )
proof end;

Lemma12: for b1, b2 being real number holds
not ( 0 < b1 & not b2 < b2 + b1 )
proof end;

theorem Th3: :: XREAL_1:3
for b1 being real number holds
not for b2 being real number holds not b1 < b2
proof end;

theorem Th4: :: XREAL_1:4
for b1 being real number holds
not for b2 being real number holds not b2 < b1
proof end;

theorem Th5: :: XREAL_1:5
for b1, b2 being real number holds
ex b3 being real number st
( b1 < b3 & b2 < b3 )
proof end;

Lemma13: for b1, b2, b3 being real number holds
( b1 + b2 <= b3 + b2 implies b1 <= b3 )
proof end;

theorem Th6: :: XREAL_1:6
for b1, b2 being real number holds
ex b3 being real number st
( b3 < b1 & b3 < b2 )
proof end;

Lemma14: for b1, b2 being Element of REAL
for b3, b4 being real number holds
( b1 = b3 & b2 = b4 implies * b1,b2 = b3 * b4 )
proof end;

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 )
proof end;

Lemma16: for b1, b2, b3 being real number holds
not ( 0 < b1 & b2 < b3 & not b2 * b1 < b3 * b1 )
proof end;

theorem Th7: :: XREAL_1:7
for b1, b2 being real number holds
not ( b1 < b2 & ( for b3 being real number holds
not ( b1 < b3 & b3 < b2 ) ) )
proof end;

theorem Th8: :: XREAL_1:8
for b1, b2, b3 being real number holds
( b1 <= b2 iff b1 + b3 <= b2 + b3 ) by Lemma8, Lemma13;

theorem Th9: :: XREAL_1:9
for b1, b2, b3, b4 being real number holds
( b1 <= b2 & b3 <= b4 implies b1 + b3 <= b2 + b4 ) by Lemma9;

theorem Th10: :: XREAL_1:10
for b1, b2, b3, b4 being real number holds
not ( b1 < b2 & b3 <= b4 & not b1 + b3 < b2 + b4 ) by Lemma11;

Lemma18: for b1, b2 being real number holds
( b1 <= b2 implies - b2 <= - b1 )
proof end;

Lemma19: for b1, b2 being real number holds
( - b1 <= - b2 implies b2 <= b1 )
proof end;

theorem Th11: :: XREAL_1:11
for b1, b2, b3 being real number holds
( b1 <= b2 iff b1 - b3 <= b2 - b3 )
proof end;

theorem Th12: :: XREAL_1:12
for b1, b2, b3 being real number holds
( b1 <= b2 iff b3 - b2 <= b3 - b1 )
proof end;

Lemma21: for b1, b2, b3 being real number holds
( b1 + b2 <= b3 implies b1 <= b3 - b2 )
proof end;

Lemma22: for b1, b2, b3 being real number holds
( b1 <= b2 - b3 implies b1 + b3 <= b2 )
proof end;

Lemma23: for b1, b2, b3 being real number holds
( b1 <= b2 + b3 implies b1 - b2 <= b3 )
proof end;

Lemma24: for b1, b2, b3 being real number holds
( b1 - b2 <= b3 implies b1 <= b2 + b3 )
proof end;

theorem Th13: :: XREAL_1:13
for b1, b2, b3 being real number holds
( b1 <= b2 - b3 implies b3 <= b2 - b1 )
proof end;

theorem Th14: :: XREAL_1:14
for b1, b2, b3 being real number holds
( b1 - b2 <= b3 implies b1 - b3 <= b2 )
proof end;

theorem Th15: :: XREAL_1:15
for b1, b2, b3, b4 being real number holds
( b1 <= b2 & b3 <= b4 implies b1 - b4 <= b2 - b3 )
proof end;

theorem Th16: :: XREAL_1:16
for b1, b2, b3, b4 being real number holds
not ( b1 < b2 & b3 <= b4 & not b1 - b4 < b2 - b3 )
proof end;

theorem Th17: :: XREAL_1:17
for b1, b2, b3, b4 being real number holds
not ( b1 <= b2 & b3 < b4 & not b1 - b4 < b2 - b3 )
proof end;

Lemma27: for b1, b2, b3, b4 being real number holds
( b1 + b2 <= b3 + b4 implies b1 - b3 <= b4 - b2 )
proof end;

theorem Th18: :: XREAL_1:18
for b1, b2, b3, b4 being real number holds
( b1 - b2 <= b3 - b4 implies b1 - b3 <= b2 - b4 )
proof end;

theorem Th19: :: XREAL_1:19
for b1, b2, b3, b4 being real number holds
( b1 - b2 <= b3 - b4 implies b4 - b2 <= b3 - b1 )
proof end;

theorem Th20: :: XREAL_1:20
for b1, b2, b3, b4 being real number holds
( b1 - b2 <= b3 - b4 implies b4 - b3 <= b2 - b1 )
proof end;

theorem Th21: :: XREAL_1:21
for b1, b2, b3 being real number holds
( b1 + b2 <= b3 iff b1 <= b3 - b2 ) by Lemma21, Lemma22;

theorem Th22: :: XREAL_1:22
for b1, b2, b3 being real number holds
( b1 <= b2 + b3 iff b1 - b2 <= b3 ) by Lemma23, Lemma24;

theorem Th23: :: XREAL_1:23
for b1, b2, b3, b4 being real number holds
( b1 + b2 <= b3 + b4 iff b1 - b3 <= b4 - b2 )
proof end;

theorem Th24: :: XREAL_1:24
for b1, b2, b3, b4 being real number holds
( b1 + b2 <= b3 - b4 implies b1 + b4 <= b3 - b2 )
proof end;

theorem Th25: :: XREAL_1:25
for b1, b2, b3, b4 being real number holds
( b1 - b2 <= b3 + b4 implies b1 - b4 <= b3 + b2 )
proof end;

theorem Th26: :: XREAL_1:26
for b1, b2 being real number holds
( b1 <= b2 iff - b2 <= - b1 ) by Lemma18, Lemma19;

Lemma28: for b1, b2 being real number holds
not ( b1 < b2 & not 0 < b2 - b1 )
proof end;

theorem Th27: :: XREAL_1:27
for b1, b2 being real number holds
( b1 <= - b2 implies b2 <= - b1 )
proof end;

Lemma30: for b1, b2 being real number holds
( b1 <= b2 implies 0 <= b2 - b1 )
proof end;

theorem Th28: :: XREAL_1:28
for b1, b2 being real number holds
( - b1 <= b2 implies - b2 <= b1 )
proof end;

theorem Th29: :: XREAL_1:29
for b1, b2 being real number holds
( 0 <= b1 & 0 <= b2 & b1 + b2 = 0 implies b1 = 0 )
proof end;

theorem Th30: :: XREAL_1:30
for b1, b2 being real number holds
( b1 <= 0 & b2 <= 0 & b1 + b2 = 0 implies b1 = 0 )
proof end;

theorem Th31: :: XREAL_1:31
for b1, b2 being real number holds
not ( 0 < b1 & not b2 < b2 + b1 ) by Lemma12;

theorem Th32: :: XREAL_1:32
for b1, b2 being real number holds
not ( b1 < 0 & not b1 + b2 < b2 )
proof end;

Lemma33: for b1, b2 being real number holds
not ( b1 < b2 & not b1 - b2 < 0 )
proof end;

Lemma34: for b1 being real number holds
( not ( b1 < 0 & not 0 < - b1 ) & not ( 0 < - b1 & not b1 < 0 ) )
proof end;

theorem Th33: :: XREAL_1:33
for b1, b2 being real number holds
( 0 <= b1 implies b2 <= b1 + b2 )
proof end;

theorem Th34: :: XREAL_1:34
for b1, b2 being real number holds
( b1 <= 0 implies b1 + b2 <= b2 )
proof end;

theorem Th35: :: XREAL_1:35
for b1, b2 being real number holds
( 0 <= b1 & 0 <= b2 implies 0 <= b1 + b2 )
proof end;

theorem Th36: :: XREAL_1:36
for b1, b2 being real number holds
not ( 0 <= b1 & 0 < b2 & not 0 < b1 + b2 )
proof end;

theorem Th37: :: XREAL_1:37
for b1, b2, b3 being real number holds
( b1 <= 0 & b2 <= b3 implies b2 + b1 <= b3 )
proof end;

theorem Th38: :: XREAL_1:38
for b1, b2, b3 being real number holds
not ( b1 <= 0 & b2 < b3 & not b2 + b1 < b3 )
proof end;

theorem Th39: :: XREAL_1:39
for b1, b2, b3 being real number holds
not ( b1 < 0 & b2 <= b3 & not b2 + b1 < b3 )
proof end;

theorem Th40: :: XREAL_1:40
for b1, b2, b3 being real number holds
( 0 <= b1 & b2 <= b3 implies b2 <= b1 + b3 )
proof end;

theorem Th41: :: XREAL_1:41
for b1, b2, b3 being real number holds
not ( 0 < b1 & b2 <= b3 & not b2 < b1 + b3 )
proof end;

theorem Th42: :: XREAL_1:42
for b1, b2, b3 being real number holds
not ( 0 <= b1 & b2 < b3 & not b2 < b1 + b3 )
proof end;

Lemma39: for b1, b2, b3 being real number holds
not ( b1 < 0 & b2 < b3 & not b3 * b1 < b2 * b1 )
proof end;

Lemma40: for b1 being real number holds
not ( 0 < b1 & not 0 < b1 " )
proof end;

Lemma41: for b1, b2, b3 being real number holds
( 0 <= b1 & b2 <= b3 implies b2 / b1 <= b3 / b1 )
proof end;

Lemma42: for b1, b2, b3 being real number holds
not ( 0 < b1 & b2 < b3 & not b2 / b1 < b3 / b1 )
proof end;

Lemma43: for b1 being real number holds
not ( 0 < b1 & not 0 < b1 / 2 )
proof end;

Lemma44: for b1 being real number holds
not ( 0 < b1 & not b1 / 2 < b1 )
proof end;

theorem Th43: :: XREAL_1:43
for b1, b2 being real number holds
( ( for b3 being real number holds
( b3 > 0 implies b1 <= b2 + b3 ) ) implies b1 <= b2 )
proof end;

theorem Th44: :: XREAL_1:44
for b1, b2 being real number holds
( ( for b3 being real number holds
( b3 < 0 implies b1 + b3 <= b2 ) ) implies b1 <= b2 )
proof end;

theorem Th45: :: XREAL_1:45
for b1, b2 being real number holds
( 0 <= b1 implies b2 - b1 <= b2 )
proof end;

theorem Th46: :: XREAL_1:46
for b1, b2 being real number holds
not ( 0 < b1 & not b2 - b1 < b2 )
proof end;

theorem Th47: :: XREAL_1:47
for b1, b2 being real number holds
( b1 <= 0 implies b2 <= b2 - b1 )
proof end;

theorem Th48: :: XREAL_1:48
for b1, b2 being real number holds
not ( b1 < 0 & not b2 < b2 - b1 )
proof end;

theorem Th49: :: XREAL_1:49
for b1, b2 being real number holds
( b1 <= b2 implies b1 - b2 <= 0 )
proof end;

theorem Th50: :: XREAL_1:50
for b1, b2 being real number holds
( b1 <= b2 implies 0 <= b2 - b1 )
proof end;

theorem Th51: :: XREAL_1:51
for b1, b2 being real number holds
not ( b1 < b2 & not b1 - b2 < 0 ) by Lemma33;

theorem Th52: :: XREAL_1:52
for b1, b2 being real number holds
not ( b1 < b2 & not 0 < b2 - b1 ) by Lemma28;

theorem Th53: :: XREAL_1:53
for b1, b2, b3 being real number holds
not ( 0 <= b1 & b2 < b3 & not b2 - b1 < b3 )
proof end;

theorem Th54: :: XREAL_1:54
for b1, b2, b3 being real number holds
( b1 <= 0 & b2 <= b3 implies b2 <= b3 - b1 )
proof end;

theorem Th55: :: XREAL_1:55
for b1, b2, b3 being real number holds
not ( b1 <= 0 & b2 < b3 & not b2 < b3 - b1 )
proof end;

theorem Th56: :: XREAL_1:56
for b1, b2, b3 being real number holds
not ( b1 < 0 & b2 <= b3 & not b2 < b3 - b1 )
proof end;

theorem Th57: :: XREAL_1:57
for b1, b2 being real number holds
not ( b1 <> b2 & not 0 < b1 - b2 & not 0 < b2 - b1 )
proof end;

theorem Th58: :: XREAL_1:58
for b1, b2 being real number holds
( ( for b3 being real number holds
( b3 < 0 implies b1 <= b2 - b3 ) ) implies b2 >= b1 )
proof end;

theorem Th59: :: XREAL_1:59
for b1, b2 being real number holds
( ( for b3 being real number holds
( b3 > 0 implies b1 - b3 <= b2 ) ) implies b1 <= b2 )
proof end;

theorem Th60: :: XREAL_1:60
for b1 being real number holds
( not ( b1 < 0 & not 0 < - b1 ) & not ( 0 < - b1 & not b1 < 0 ) ) by Lemma34;

theorem Th61: :: XREAL_1:61
for b1, b2 being real number holds
( b1 <= - b2 implies b1 + b2 <= 0 )
proof end;

theorem Th62: :: XREAL_1:62
for b1, b2 being real number holds
( - b1 <= b2 implies 0 <= b1 + b2 )
proof end;

theorem Th63: :: XREAL_1:63
for b1, b2 being real number holds
not ( b1 < - b2 & not b1 + b2 < 0 )
proof end;

theorem Th64: :: XREAL_1:64
for b1, b2 being real number holds
not ( - b1 < b2 & not 0 < b2 + b1 )
proof end;

Lemma46: for b1, b2, b3 being real number holds
( b1 <= b2 & b3 <= 0 implies b2 * b3 <= b1 * b3 )
proof end;

theorem Th65: :: XREAL_1:65
for b1 being real number holds 0 <= b1 * b1
proof end;

theorem Th66: :: XREAL_1:66
for b1, b2, b3 being real number holds
( b1 <= b2 & 0 <= b3 implies b1 * b3 <= b2 * b3 ) by Lemma15;

theorem Th67: :: XREAL_1:67
for b1, b2, b3 being real number holds
( b1 <= b2 & b3 <= 0 implies b2 * b3 <= b1 * b3 ) by Lemma46;

theorem Th68: :: XREAL_1:68
for b1, b2, b3, b4 being real number holds
( 0 <= b1 & b1 <= b2 & 0 <= b3 & b3 <= b4 implies b1 * b3 <= b2 * b4 )
proof end;

theorem Th69: :: XREAL_1:69
for b1, b2, b3, b4 being real number holds
( b1 <= 0 & b2 <= b1 & b3 <= 0 & b4 <= b3 implies b1 * b3 <= b2 * b4 )
proof end;

theorem Th70: :: XREAL_1:70
for b1, b2, b3 being real number holds
not ( 0 < b1 & b2 < b3 & not b2 * b1 < b3 * b1 ) by Lemma16;

theorem Th71: :: XREAL_1:71
for b1, b2, b3 being real number holds
not ( b1 < 0 & b2 < b3 & not b3 * b1 < b2 * b1 ) by Lemma39;

theorem Th72: :: XREAL_1:72
for b1, b2, b3, b4 being real number holds
not ( b1 < 0 & b2 <= b1 & b3 < 0 & b4 < b3 & not b1 * b3 < b2 * b4 )
proof end;

theorem Th73: :: XREAL_1:73
for b1, b2, b3, b4 being real number holds
not ( 0 <= b1 & 0 <= b2 & 0 <= b3 & 0 <= b4 & (b1 * b3) + (b2 * b4) = 0 & not b1 = 0 & not b3 = 0 )
proof end;

Lemma49: for b1, b2, b3 being real number holds
not ( b1 < 0 & b2 < b3 & not b3 / b1 < b2 / b1 )
proof end;

Lemma50: for b1, b2, b3 being real number holds
not ( b1 <= 0 & b2 / b1 < b3 / b1 & not b3 < b2 )
proof end;

theorem Th74: :: XREAL_1:74
for b1, b2, b3 being real number holds
( 0 <= b1 & b2 <= b3 implies b2 / b1 <= b3 / b1 ) by Lemma41;

theorem Th75: :: XREAL_1:75
for b1, b2, b3 being real number holds
( b1 <= 0 & b2 <= b3 implies b3 / b1 <= b2 / b1 ) by Lemma50;

theorem Th76: :: XREAL_1:76
for b1, b2, b3 being real number holds
not ( 0 < b1 & b2 < b3 & not b2 / b1 < b3 / b1 ) by Lemma42;

theorem Th77: :: XREAL_1:77
for b1, b2, b3 being real number holds
not ( b1 < 0 & b2 < b3 & not b3 / b1 < b2 / b1 ) by Lemma49;

theorem Th78: :: XREAL_1:78
for b1, b2, b3 being real number holds
not ( 0 < b1 & 0 < b2 & b2 < b3 & not b1 / b3 < b1 / b2 )
proof end;

Lemma51: for b1, b2 being real number holds
not ( b1 < 0 & 0 < b2 & not b2 / b1 < 0 )
proof end;

Lemma52: for b1, b2 being real number holds
not ( b1 < 0 & b2 < b1 & not b1 " < b2 " )
proof end;

theorem Th79: :: XREAL_1:79
for b1, b2, b3 being real number holds
( 0 < b1 & b2 * b1 <= b3 implies b2 <= b3 / b1 )
proof end;

theorem Th80: :: XREAL_1:80
for b1, b2, b3 being real number holds
( b1 < 0 & b2 * b1 <= b3 implies b3 / b1 <= b2 )
proof end;

theorem Th81: :: XREAL_1:81
for b1, b2, b3 being real number holds
( 0 < b1 & b2 <= b3 * b1 implies b2 / b1 <= b3 )
proof end;

theorem Th82: :: XREAL_1:82
for b1, b2, b3 being real number holds
( b1 < 0 & b2 <= b3 * b1 implies b3 <= b2 / b1 )
proof end;

theorem Th83: :: XREAL_1:83
for b1, b2, b3 being real number holds
not ( 0 < b1 & b2 * b1 < b3 & not b2 < b3 / b1 )
proof end;

theorem Th84: :: XREAL_1:84
for b1, b2, b3 being real number holds
not ( b1 < 0 & b2 * b1 < b3 & not b3 / b1 < b2 )
proof end;

theorem Th85: :: XREAL_1:85
for b1, b2, b3 being real number holds
not ( 0 < b1 & b2 < b3 * b1 & not b2 / b1 < b3 )
proof end;

theorem Th86: :: XREAL_1:86
for b1, b2, b3 being real number holds
not ( b1 < 0 & b2 < b3 * b1 & not b3 < b2 / b1 )
proof end;

Lemma61: for b1, b2 being real number holds
( 0 < b1 & b1 <= b2 implies b2 " <= b1 " )
proof end;

Lemma62: for b1, b2 being real number holds
( b1 < 0 & b2 <= b1 implies b1 " <= b2 " )
proof end;

theorem Th87: :: XREAL_1:87
for b1, b2 being real number holds
( 0 < b1 & b1 <= b2 implies b2 " <= b1 " ) by Lemma61;

theorem Th88: :: XREAL_1:88
for b1, b2 being real number holds
( b1 < 0 & b2 <= b1 implies b1 " <= b2 " ) by Lemma62;

theorem Th89: :: XREAL_1:89
for b1, b2 being real number holds
not ( b1 < 0 & b2 < b1 & not b1 " < b2 " ) by Lemma52;

Lemma63: for b1, b2 being real number holds
not ( 0 < b1 & 0 < b2 & not 0 < b1 / b2 )
proof end;

Lemma64: for b1, b2 being real number holds
not ( 0 < b1 & b1 < b2 & not b2 " < b1 " )
proof end;

theorem Th90: :: XREAL_1:90
for b1, b2 being real number holds
not ( 0 < b1 & b1 < b2 & not b2 " < b1 " ) by Lemma64;

theorem Th91: :: XREAL_1:91
for b1, b2 being real number holds
( 0 < b1 " & b1 " <= b2 " implies b2 <= b1 )
proof end;

theorem Th92: :: XREAL_1:92
for b1, b2 being real number holds
( b1 " < 0 & b2 " <= b1 " implies b1 <= b2 )
proof end;

theorem Th93: :: XREAL_1:93
for b1, b2 being real number holds
not ( 0 < b1 " & b1 " < b2 " & not b2 < b1 )
proof end;

theorem Th94: :: XREAL_1:94
for b1, b2 being real number holds
not ( b1 " < 0 & b2 " < b1 " & not b1 < b2 )
proof end;

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
for b1, b2 being real number holds
( 0 <= b1 & (b2 - b1) * (b2 + b1) <= 0 implies ( - b1 <= b2 & b2 <= b1 ) )
proof end;

theorem Th96: :: XREAL_1:96
for b1, b2 being real number holds
( 0 <= b1 & (b2 - b1) * (b2 + b1) < 0 implies ( - b1 < b2 & b2 < b1 ) )
proof end;

theorem Th97: :: XREAL_1:97
for b1, b2 being real number holds
not ( 0 <= b1 & 0 <= (b2 - b1) * (b2 + b1) & not b2 <= - b1 & not b1 <= b2 )
proof end;

theorem Th98: :: XREAL_1:98
for b1, b2, b3, b4 being real number holds
not ( 0 <= b1 & 0 <= b2 & b1 < b3 & b2 < b4 & not b1 * b2 < b3 * b4 )
proof end;

theorem Th99: :: XREAL_1:99
for b1, b2, b3, b4 being real number holds
not ( 0 <= b1 & 0 < b2 & b1 < b3 & b2 <= b4 & not b1 * b2 < b3 * b4 )
proof end;

theorem Th100: :: XREAL_1:100
for b1, b2, b3, b4 being real number holds
not ( 0 < b1 & 0 < b2 & b1 <= b3 & b2 < b4 & not b1 * b2 < b3 * b4 )
proof end;

theorem Th101: :: XREAL_1:101
for b1, b2, b3 being real number holds
not ( 0 < b1 & b2 < 0 & b3 < b2 & not b1 / b2 < b1 / b3 )
proof end;

theorem Th102: :: XREAL_1:102
for b1, b2, b3 being real number holds
not ( b1 < 0 & 0 < b2 & b2 < b3 & not b1 / b2 < b1 / b3 )
proof end;

theorem Th103: :: XREAL_1:103
for b1, b2, b3 being real number holds
not ( b1 < 0 & b2 < 0 & b3 < b2 & not b1 / b3 < b1 / b2 )
proof end;

theorem Th104: :: XREAL_1:104
for b1, b2, b3, b4 being real number holds
( 0 < b1 & 0 < b2 & b3 * b2 <= b4 * b1 implies b3 / b1 <= b4 / b2 )
proof end;

theorem Th105: :: XREAL_1:105
for b1, b2, b3, b4 being real number holds
( b1 < 0 & b2 < 0 & b3 * b2 <= b4 * b1 implies b3 / b1 <= b4 / b2 )
proof end;

theorem Th106: :: XREAL_1:106
for b1, b2, b3, b4 being real number holds
( 0 < b1 & b2 < 0 & b3 * b2 <= b4 * b1 implies b4 / b2 <= b3 / b1 )
proof end;

theorem Th107: :: XREAL_1:107
for b1, b2, b3, b4 being real number holds
( b1 < 0 & 0 < b2 & b3 * b2 <= b4 * b1 implies b4 / b2 <= b3 / b1 )
proof end;

theorem Th108: :: XREAL_1:108
for b1, b2, b3, b4 being real number holds
not ( 0 < b1 & 0 < b2 & b3 * b2 < b4 * b1 & not b3 / b1 < b4 / b2 )
proof end;

theorem Th109: :: XREAL_1:109
for b1, b2, b3, b4 being real number holds
not ( b1 < 0 & b2 < 0 & b3 * b2 < b4 * b1 & not b3 / b1 < b4 / b2 )
proof end;

theorem Th110: :: XREAL_1:110
for b1, b2, b3, b4 being real number holds
not ( 0 < b1 & b2 < 0 & b3 * b2 < b4 * b1 & not b4 / b2 < b3 / b1 )
proof end;

theorem Th111: :: XREAL_1:111
for b1, b2, b3, b4 being real number holds
not ( b1 < 0 & 0 < b2 & b3 * b2 < b4 * b1 & not b4 / b2 < b3 / b1 )
proof end;

theorem Th112: :: XREAL_1:112
for b1, b2, b3, b4 being real number holds
not ( b1 < 0 & b2 < 0 & b3 * b1 < b4 / b2 & not b3 * b2 < b4 / b1 )
proof end;

theorem Th113: :: XREAL_1:113
for b1, b2, b3, b4 being real number holds
not ( 0 < b1 & 0 < b2 & b3 * b1 < b4 / b2 & not b3 * b2 < b4 / b1 )
proof end;

theorem Th114: :: XREAL_1:114
for b1, b2, b3, b4 being real number holds
not ( b1 < 0 & b2 < 0 & b3 / b2 < b4 * b1 & not b3 / b1 < b4 * b2 )
proof end;

theorem Th115: :: XREAL_1:115
for b1, b2, b3, b4 being real number holds
not ( 0 < b1 & 0 < b2 & b3 / b2 < b4 * b1 & not b3 / b1 < b4 * b2 )
proof end;

theorem Th116: :: XREAL_1:116
for b1, b2, b3, b4 being real number holds
not ( b1 < 0 & 0 < b2 & b3 * b1 < b4 / b2 & not b4 / b1 < b3 * b2 )
proof end;

theorem Th117: :: XREAL_1:117
for b1, b2, b3, b4 being real number holds
not ( 0 < b1 & b2 < 0 & b3 * b1 < b4 / b2 & not b4 / b1 < b3 * b2 )
proof end;

theorem Th118: :: XREAL_1:118
for b1, b2, b3, b4 being real number holds
not ( b1 < 0 & 0 < b2 & b3 / b2 < b4 * b1 & not b4 * b2 < b3 / b1 )
proof end;

theorem Th119: :: XREAL_1:119
for b1, b2, b3, b4 being real number holds
not ( 0 < b1 & b2 < 0 & b3 / b2 < b4 * b1 & not b4 * b2 < b3 / b1 )
proof end;

theorem Th120: :: XREAL_1:120
for b1, b2, b3 being real number holds
( 0 < b1 & 0 <= b2 & b1 <= b3 implies b2 / b3 <= b2 / b1 )
proof end;

theorem Th121: :: XREAL_1:121
for b1, b2, b3 being real number holds
( 0 <= b1 & b2 < 0 & b3 <= b2 implies b1 / b2 <= b1 / b3 )
proof end;

theorem Th122: :: XREAL_1:122
for b1, b2, b3 being real number holds
( b1 <= 0 & 0 < b2 & b2 <= b3 implies b1 / b2 <= b1 / b3 )
proof end;

theorem Th123: :: XREAL_1:123
for b1, b2, b3 being real number holds
( b1 <= 0 & b2 < 0 & b3 <= b2 implies b1 / b3 <= b1 / b2 )
proof end;

theorem Th124: :: XREAL_1:124
for b1 being real number holds
( not ( 0 < b1 & not 0 < b1 " ) & not ( 0 < b1 " & not 0 < b1 ) )
proof end;

theorem Th125: :: XREAL_1:125
for b1 being real number holds
( not ( b1 < 0 & not b1 " < 0 ) & not ( b1 " < 0 & not b1 < 0 ) )
proof end;

theorem Th126: :: XREAL_1:126
for b1, b2 being real number holds
not ( b1 < 0 & 0 < b2 & not b1 " < b2 " )
proof end;

theorem Th127: :: XREAL_1:127
for b1, b2 being real number holds
not ( b1 " < 0 & 0 < b2 " & not b1 < b2 )
proof end;

theorem Th128: :: XREAL_1:128
for b1, b2 being real number holds
( b1 <= 0 & 0 <= b1 implies 0 = b1 * b2 ) by Lemma69;

theorem Th129: :: XREAL_1:129
for b1, b2 being real number holds
( 0 <= b1 & 0 <= b2 implies 0 <= b1 * b2 ) by Lemma67;

theorem Th130: :: XREAL_1:130
for b1, b2 being real number holds
( b1 <= 0 & b2 <= 0 implies 0 <= b1 * b2 ) by Lemma68;

theorem Th131: :: XREAL_1:131
for b1, b2 being real number holds
not ( 0 < b1 & 0 < b2 & not 0 < b1 * b2 ) by Lemma65;

theorem Th132: :: XREAL_1:132
for b1, b2 being real number holds
not ( b1 < 0 & b2 < 0 & not 0 < b1 * b2 ) by Lemma66;

theorem Th133: :: XREAL_1:133
for b1, b2 being real number holds
( 0 <= b1 & b2 <= 0 implies b1 * b2 <= 0 )
proof end;

theorem Th134: :: XREAL_1:134
for b1, b2 being real number holds
not ( 0 < b1 & b2 < 0 & not b1 * b2 < 0 ) by Lemma70;

theorem Th135: :: XREAL_1:135
for b1, b2 being real number holds
not ( b1 * b2 < 0 & not ( b1 > 0 & b2 < 0 ) & not ( b1 < 0 & b2 > 0 ) )
proof end;

theorem Th136: :: XREAL_1:136
for b1, b2 being real number holds
not ( b1 * b2 > 0 & not ( b1 > 0 & b2 > 0 ) & not ( b1 < 0 & b2 < 0 ) )
proof end;

theorem Th137: :: XREAL_1:137
for b1, b2 being real number holds
( b1 <= 0 & b2 <= 0 implies 0 <= b1 / b2 )
proof end;

theorem Th138: :: XREAL_1:138
for b1, b2 being real number holds
( 0 <= b1 & 0 <= b2 implies 0 <= b1 / b2 )
proof end;

theorem Th139: :: XREAL_1:139
for b1, b2 being real number holds
( 0 <= b1 & b2 <= 0 implies b1 / b2 <= 0 )
proof end;

theorem Th140: :: XREAL_1:140
for b1, b2 being real number holds
( b1 <= 0 & 0 <= b2 implies b1 / b2 <= 0 )
proof end;

theorem Th141: :: XREAL_1:141
for b1, b2 being real number holds
not ( 0 < b1 & 0 < b2 & not 0 < b1 / b2 ) by Lemma63;

theorem Th142: :: XREAL_1:142
for b1, b2 being real number holds
not ( b1 < 0 & b2 < 0 & not 0 < b1 / b2 )
proof end;

theorem Th143: :: XREAL_1:143
for b1, b2 being real number holds
not ( b1 < 0 & 0 < b2 & not b1 / b2 < 0 )
proof end;

theorem Th144: :: XREAL_1:144
for b1, b2 being real number holds
not ( b1 < 0 & 0 < b2 & not b2 / b1 < 0 ) by Lemma51;

theorem Th145: :: XREAL_1:145
for b1, b2 being real number holds
not ( b1 / b2 < 0 & not ( b2 < 0 & b1 > 0 ) & not ( b2 > 0 & b1 < 0 ) )
proof end;

theorem Th146: :: XREAL_1:146
for b1, b2 being real number holds
not ( b1 / b2 > 0 & not ( b2 > 0 & b1 > 0 ) & not ( b2 < 0 & b1 < 0 ) )
proof end;

theorem Th147: :: XREAL_1:147
for b1, b2 being real number holds
not ( b1 <= b2 & not b1 < b2 + 1 )
proof end;

theorem Th148: :: XREAL_1:148
for b1 being real number holds
b1 - 1 < b1
proof end;

theorem Th149: :: XREAL_1:149
for b1, b2 being real number holds
not ( b1 <= b2 & not b1 - 1 < b2 )
proof end;

theorem Th150: :: XREAL_1:150
for b1 being real number holds
not ( - 1 < b1 & not 0 < 1 + b1 )
proof end;

theorem Th151: :: XREAL_1:151
for b1 being real number holds
not ( b1 < 1 & not 1 - b1 > 0 ) by Lemma28;

theorem Th152: :: XREAL_1:152
for b1, b2 being real number holds
( 0 <= b1 & b1 <= 1 & 0 <= b2 & b2 <= 1 & b1 * b2 = 1 implies b1 = 1 )
proof end;

theorem Th153: :: XREAL_1:153
for b1, b2 being real number holds
( 0 <= b1 & 1 <= b2 implies b1 <= b1 * b2 )
proof end;

theorem Th154: :: XREAL_1:154
for b1, b2 being real number holds
( b1 <= 0 & b2 <= 1 implies b1 <= b1 * b2 )
proof end;

theorem Th155: :: XREAL_1:155
for b1, b2 being real number holds
( 0 <= b1 & b2 <= 1 implies b1 * b2 <= b1 )
proof end;

theorem Th156: :: XREAL_1:156
for b1, b2 being real number holds
( b1 <= 0 & 1 <= b2 implies b1 * b2 <= b1 )
proof end;

theorem Th157: :: XREAL_1:157
for b1, b2 being real number holds
not ( 0 < b1 & 1 < b2 & not b1 < b1 * b2 )
proof end;

theorem Th158: :: XREAL_1:158
for b1, b2 being real number holds
not ( b1 < 0 & b2 < 1 & not b1 < b1 * b2 )
proof end;

theorem Th159: :: XREAL_1:159
for b1, b2 being real number holds
not ( 0 < b1 & b2 < 1 & not b1 * b2 < b1 )
proof end;

theorem Th160: :: XREAL_1:160
for b1, b2 being real number holds
not ( b1 < 0 & 1 < b2 & not b1 * b2 < b1 )
proof end;

theorem Th161: :: XREAL_1:161
for b1, b2 being real number holds
( 1 <= b1 & 1 <= b2 implies 1 <= b1 * b2 )
proof end;

theorem Th162: :: XREAL_1:162
for b1, b2 being real number holds
( 0 <= b1 & b1 <= 1 & 0 <= b2 & b2 <= 1 implies b1 * b2 <= 1 )
proof end;

theorem Th163: :: XREAL_1:163
for b1, b2 being real number holds
not ( 1 < b1 & 1 <= b2 & not 1 < b1 * b2 )
proof end;

theorem Th164: :: XREAL_1:164
for b1, b2 being real number holds
not ( 0 <= b1 & b1 < 1 & 0 <= b2 & b2 <= 1 & not b1 * b2 < 1 )
proof end;

theorem Th165: :: XREAL_1:165
for b1, b2 being real number holds
( b1 <= - 1 & b2 <= - 1 implies 1 <= b1 * b2 )
proof end;

theorem Th166: :: XREAL_1:166
for b1, b2 being real number holds
not ( b1 < - 1 & b2 <= - 1 & not 1 < b1 * b2 )
proof end;

theorem Th167: :: XREAL_1:167
for b1, b2 being real number holds
( b1 <= 0 & - 1 <= b1 & b2 <= 0 & - 1 <= b2 implies b1 * b2 <= 1 )
proof end;

theorem Th168: :: XREAL_1:168
for b1, b2 being real number holds
not ( b1 <= 0 & - 1 < b1 & b2 <= 0 & - 1 <= b2 & not b1 * b2 < 1 )
proof end;

theorem Th169: :: XREAL_1:169
for b1, b2 being real number holds
( ( for b3 being real number holds
( 1 < b3 implies b1 <= b2 * b3 ) ) implies b1 <= b2 )
proof end;

Lemma80: for b1 being real number holds
not ( 1 < b1 & not b1 " < 1 )
proof end;

theorem Th170: :: XREAL_1:170
for b1, b2 being real number holds
( ( for b3 being real number holds
( 0 < b3 & b3 < 1 implies b1 * b3 <= b2 ) ) implies b1 <= b2 )
proof end;

Lemma81: for b1, b2 being real number holds
( b1 <= b2 & 0 <= b1 implies b1 / b2 <= 1 )
proof end;

Lemma82: for b1, b2 being real number holds
( b1 <= b2 & 0 <= b2 implies b1 / b2 <= 1 )
proof end;

theorem Th171: :: XREAL_1:171
for b1, b2 being real number holds
( ( for b3 being real number holds
( 0 < b3 & b3 < 1 implies b1 <= b3 * b2 ) ) implies b1 <= 0 )
proof end;

theorem Th172: :: XREAL_1:172
for b1, b2, b3 being real number holds
not ( 0 <= b1 & b1 <= 1 & 0 <= b2 & 0 <= b3 & (b1 * b2) + ((1 - b1) * b3) = 0 & not ( b1 = 0 & b3 = 0 ) & not ( b1 = 1 & b2 = 0 ) & not ( b2 = 0 & b3 = 0 ) )
proof end;

theorem Th173: :: XREAL_1:173
for b1, b2, b3 being real number holds
( 0 <= b1 & b1 <= 1 & b2 <= b3 implies b2 <= ((1 - b1) * b2) + (b1 * b3) )
proof end;

theorem Th174: :: XREAL_1:174
for b1, b2, b3 being real number holds
( 0 <= b1 & b1 <= 1 & b2 <= b3 implies ((1 - b1) * b2) + (b1 * b3) <= b3 )
proof end;

theorem Th175: :: XREAL_1:175
for b1, b2, b3, b4 being real number holds
( 0 <= b1 & b1 <= 1 & b2 <= b3 & b2 <= b4 implies b2 <= ((1 - b1) * b3) + (b1 * b4) )
proof end;

theorem Th176: :: XREAL_1:176
for b1, b2, b3, b4 being real number holds
( 0 <= b1 & b1 <= 1 & b2 <= b3 & b4 <= b3 implies ((1 - b1) * b2) + (b1 * b4) <= b3 )
proof end;

theorem Th177: :: XREAL_1:177
for b1, b2, b3, b4 being real number holds
not ( 0 <= b1 & b1 <= 1 & b2 < b3 & b2 < b4 & not b2 < ((1 - b1) * b3) + (b1 * b4) )
proof end;

theorem Th178: :: XREAL_1:178
for b1, b2, b3, b4 being real number holds
not ( 0 <= b1 & b1 <= 1 & b2 < b3 & b4 < b3 & not ((1 - b1) * b2) + (b1 * b4) < b3 )
proof end;

theorem Th179: :: XREAL_1:179
for b1, b2, b3, b4 being real number holds
not ( 0 < b1 & b1 < 1 & b2 <= b3 & b2 < b4 & not b2 < ((1 - b1) * b3) + (b1 * b4) )
proof end;

theorem Th180: :: XREAL_1:180
for b1, b2, b3, b4 being real number holds
not ( 0 < b1 & b1 < 1 & b2 < b3 & b4 <= b3 & not ((1 - b1) * b2) + (b1 * b4) < b3 )
proof end;

theorem Th181: :: XREAL_1:181
for b1, b2, b3 being real number holds
( 0 <= b1 & b1 <= 1 & b2 <= ((1 - b1) * b2) + (b1 * b3) & b3 < ((1 - b1) * b2) + (b1 * b3) implies b1 = 0 )
proof end;

theorem Th182: :: XREAL_1:182
for b1, b2, b3 being real number holds
( 0 <= b1 & b1 <= 1 & ((1 - b1) * b2) + (b1 * b3) <= b2 & ((1 - b1) * b2) + (b1 * b3) < b3 implies b1 = 0 )
proof end;

theorem Th183: :: XREAL_1:183
for b1, b2 being real number holds
( 0 < b1 & b1 <= b2 implies 1 <= b2 / b1 )
proof end;

theorem Th184: :: XREAL_1:184
for b1, b2 being real number holds
( b1 < 0 & b2 <= b1 implies 1 <= b2 / b1 )
proof end;

theorem Th185: :: XREAL_1:185
for b1, b2 being real number holds
( 0 <= b1 & b1 <= b2 implies b1 / b2 <= 1 ) by Lemma81;

theorem Th186: :: XREAL_1:186
for b1, b2 being real number holds
( b1 <= b2 & b2 <= 0 implies b2 / b1 <= 1 )
proof end;

theorem Th187: :: XREAL_1:187
for b1, b2 being real number holds
( 0 <= b1 & b2 <= b1 implies b2 / b1 <= 1 ) by Lemma82;

theorem Th188: :: XREAL_1:188
for b1, b2 being real number holds
( b1 <= 0 & b1 <= b2 implies b2 / b1 <= 1 )
proof end;

theorem Th189: :: XREAL_1:189
for b1, b2 being real number holds
not ( 0 < b1 & b1 < b2 & not 1 < b2 / b1 )
proof end;

theorem Th190: :: XREAL_1:190
for b1, b2 being real number holds
not ( b1 < 0 & b2 < b1 & not 1 < b2 / b1 )
proof end;

theorem Th191: :: XREAL_1:191
for b1, b2 being real number holds
not ( 0 <= b1 & b1 < b2 & not b1 / b2 < 1 )
proof end;

theorem Th192: :: XREAL_1:192
for b1, b2 being real number holds
not ( b1 <= 0 & b2 < b1 & not b1 / b2 < 1 )
proof end;

theorem Th193: :: XREAL_1:193
for b1, b2 being real number holds
not ( 0 < b1 & b2 < b1 & not b2 / b1 < 1 )
proof end;

theorem Th194: :: XREAL_1:194
for b1, b2 being real number holds
not ( b1 < 0 & b1 < b2 & not b2 / b1 < 1 )
proof end;

theorem Th195: :: XREAL_1:195
for b1, b2 being real number holds
( 0 <= b1 & - b1 <= b2 implies - 1 <= b2 / b1 )
proof end;

theorem Th196: :: XREAL_1:196
for b1, b2 being real number holds
( 0 <= b1 & - b2 <= b1 implies - 1 <= b2 / b1 )
proof end;

theorem Th197: :: XREAL_1:197
for b1, b2 being real number holds
( b1 <= 0 & b2 <= - b1 implies - 1 <= b2 / b1 )
proof end;

theorem Th198: :: XREAL_1:198
for b1, b2 being real number holds
( b1 <= 0 & b1 <= - b2 implies - 1 <= b2 / b1 )
proof end;

theorem Th199: :: XREAL_1:199
for b1, b2 being real number holds
not ( 0 < b1 & - b1 < b2 & not - 1 < b2 / b1 )
proof end;

theorem Th200: :: XREAL_1:200
for b1, b2 being real number holds
not ( 0 < b1 & - b2 < b1 & not - 1 < b2 / b1 )
proof end;

theorem Th201: :: XREAL_1:201
for b1, b2 being real number holds
not ( b1 < 0 & b2 < - b1 & not - 1 < b2 / b1 )
proof end;

theorem Th202: :: XREAL_1:202
for b1, b2 being real number holds
not ( b1 < 0 & b1 < - b2 & not - 1 < b2 / b1 )
proof end;

theorem Th203: :: XREAL_1:203
for b1, b2 being real number holds
( 0 < b1 & b2 <= - b1 implies b2 / b1 <= - 1 )
proof end;

theorem Th204: :: XREAL_1:204
for b1, b2 being real number holds
( 0 < b1 & b1 <= - b2 implies b2 / b1 <= - 1 )
proof end;

theorem Th205: :: XREAL_1:205
for b1, b2 being real number holds
( b1 < 0 & - b1 <= b2 implies b2 / b1 <= - 1 )
proof end;

theorem Th206: :: XREAL_1:206
for b1, b2 being real number holds
( b1 < 0 & - b2 <= b1 implies b2 / b1 <= - 1 )
proof end;

theorem Th207: :: XREAL_1:207
for b1, b2 being real number holds
not ( 0 < b1 & b2 < - b1 & not b2 / b1 < - 1 )
proof end;

theorem Th208: :: XREAL_1:208
for b1, b2 being real number holds
not ( 0 < b1 & b1 < - b2 & not b2 / b1 < - 1 )
proof end;

theorem Th209: :: XREAL_1:209
for b1, b2 being real number holds
not ( b1 < 0 & - b1 < b2 & not b2 / b1 < - 1 )
proof end;

theorem Th210: :: XREAL_1:210
for b1, b2 being real number holds
not ( b1 < 0 & - b2 < b1 & not b2 / b1 < - 1 )
proof end;

theorem Th211: :: XREAL_1:211
for b1, b2 being real number holds
( ( for b3 being real number holds
( 0 < b3 & b3 < 1 implies b1 <= b2 / b3 ) ) implies b1 <= b2 )
proof end;

theorem Th212: :: XREAL_1:212
for b1, b2 being real number holds
( ( for b3 being real number holds
( 1 < b3 implies b1 / b3 <= b2 ) ) implies b1 <= b2 )
proof end;

theorem Th213: :: XREAL_1:213
for b1 being real number holds
( 1 <= b1 implies b1 " <= 1 )
proof end;

theorem Th214: :: XREAL_1:214
for b1 being real number holds
not ( 1 < b1 & not b1 " < 1 ) by Lemma80;

theorem Th215: :: XREAL_1:215
for b1 being real number holds
( b1 <= - 1 implies - 1 <= b1 " )
proof end;

theorem Th216: :: XREAL_1:216
for b1 being real number holds
not ( b1 < - 1 & not - 1 < b1 " )
proof end;

theorem Th217: :: XREAL_1:217
for b1 being real number holds
not ( 0 < b1 & not 0 < b1 / 2 ) by Lemma43;

theorem Th218: :: XREAL_1:218
for b1 being real number holds
not ( 0 < b1 & not b1 / 2 < b1 ) by Lemma44;

theorem Th219: :: XREAL_1:219
for b1 being real number holds
( b1 <= 1 / 2 implies (2 * b1) - 1 <= 0 )
proof end;

theorem Th220: :: XREAL_1:220
for b1 being real number holds
( b1 <= 1 / 2 implies 0 <= 1 - (2 * b1) )
proof end;

theorem Th221: :: XREAL_1:221
for b1 being real number holds
( b1 >= 1 / 2 implies (2 * b1) - 1 >= 0 )
proof end;

theorem Th222: :: XREAL_1:222
for b1 being real number holds
( b1 >= 1 / 2 implies 0 >= 1 - (2 * b1) )
proof end;

theorem Th223: :: XREAL_1:223
for b1 being real number holds
not ( 0 < b1 & not b1 / 3 < b1 )
proof end;

theorem Th224: :: XREAL_1:224
for b1 being real number holds
not ( 0 < b1 & not 0 < b1 / 3 )
proof end;

theorem Th225: :: XREAL_1:225
for b1 being real number holds
not ( 0 < b1 & not b1 / 4 < b1 )
proof end;

theorem Th226: :: XREAL_1:226
for b1 being real number holds
not ( 0 < b1 & not 0 < b1 / 4 )
proof end;

theorem Th227: :: XREAL_1:227
for b1, b2 being real number holds
not ( b2 <> 0 & ( for b3 being real number holds
not b1 = b2 / b3 ) )
proof end;