:: EXTREAL2 semantic presentation

theorem Th1: :: EXTREAL2:1
canceled;

theorem Th2: :: EXTREAL2:2
for b1 being R_eal holds
not ( b1 <> +infty & b1 <> -infty & ( for b2 being R_eal holds
not b1 + b2 = 0 ) )
proof end;

theorem Th3: :: EXTREAL2:3
for b1 being R_eal holds
not ( b1 <> +infty & b1 <> -infty & b1 <> 0 & ( for b2 being R_eal holds
not b1 * b2 = 1. ) )
proof end;

theorem Th4: :: EXTREAL2:4
for b1 being R_eal holds
( 1. * b1 = b1 & (R_EAL 1) * b1 = b1 )
proof end;

theorem Th5: :: EXTREAL2:5
for b1 being R_eal holds 0. - b1 = - b1
proof end;

theorem Th6: :: EXTREAL2:6
canceled;

theorem Th7: :: EXTREAL2:7
for b1, b2 being R_eal holds
( 0 <= b1 & 0 <= b2 implies 0 <= b1 + b2 )
proof end;

theorem Th8: :: EXTREAL2:8
canceled;

theorem Th9: :: EXTREAL2:9
for b1, b2 being R_eal holds
( b1 <= 0 & b2 <= 0 implies b1 + b2 <= 0 )
proof end;

theorem Th10: :: EXTREAL2:10
canceled;

theorem Th11: :: EXTREAL2:11
for b1, b2, b3 being R_eal holds
( b1 <> +infty & b1 <> -infty & b2 + b1 = b3 implies b2 = b3 - b1 )
proof end;

theorem Th12: :: EXTREAL2:12
for b1 being R_eal holds
( b1 <> +infty & b1 <> -infty & b1 <> 0 implies ( b1 * (1. / b1) = 1. & (1. / b1) * b1 = 1. ) )
proof end;

not +infty in REAL
by XXREAL_0:8;

then Lemma4: +infty + -infty = 0.
by SUPINF_2:def 2;

ex b1 being Real st
( 0. = b1 & - 0. = - b1 )
by SUPINF_2:def 3;

then Lemma5: - 0. = 0
;

Lemma6: - +infty = -infty
by SUPINF_2:def 3;

theorem Th13: :: EXTREAL2:13
for b1 being R_eal holds b1 - b1 = 0
proof end;

E7: - (+infty + -infty ) = - (+infty + -infty )
.= - 0. by Lemma4
.= 0 by Lemma5
.= +infty + -infty by Lemma4
.= +infty - +infty by SUPINF_2:def 3
.= (- -infty ) - +infty by SUPINF_2:def 3, XXREAL_0:11 ;

Lemma8: for b1 being R_eal holds
( b1 in REAL implies - (b1 + +infty ) = (- +infty ) + (- b1) )
proof end;

Lemma9: for b1 being R_eal holds
( b1 in REAL implies - (b1 + -infty ) = (- -infty ) + (- b1) )
proof end;

theorem Th14: :: EXTREAL2:14
for b1, b2 being R_eal holds - (b1 + b2) = (- b2) - b1
proof end;

theorem Th15: :: EXTREAL2:15
for b1, b2 being R_eal holds
( - (b1 - b2) = (- b1) + b2 & - (b1 - b2) = b2 - b1 )
proof end;

theorem Th16: :: EXTREAL2:16
for b1, b2 being R_eal holds
( - ((- b1) + b2) = b1 - b2 & - ((- b1) + b2) = b1 + (- b2) )
proof end;

theorem Th17: :: EXTREAL2:17
for b1, b2 being R_eal holds
( ( ( b1 = +infty & 0 < b2 & b2 < +infty ) or ( b1 = -infty & b2 < 0 & -infty < b2 ) ) implies b1 / b2 = +infty )
proof end;

theorem Th18: :: EXTREAL2:18
for b1, b2 being R_eal holds
( ( ( b1 = +infty & b2 < 0 & -infty < b2 ) or ( b1 = -infty & 0 < b2 & b2 < +infty ) ) implies b1 / b2 = -infty )
proof end;

theorem Th19: :: EXTREAL2:19
for b1, b2 being R_eal holds
( -infty < b1 & b1 < +infty & b1 <> 0 implies ( (b2 * b1) / b1 = b2 & b2 * (b1 / b1) = b2 ) )
proof end;

theorem Th20: :: EXTREAL2:20
( 1. < +infty & -infty < 1. ) by MESFUNC1:def 8, SUPINF_1:31;

theorem Th21: :: EXTREAL2:21
for b1 being R_eal holds
( ( b1 = +infty or b1 = -infty ) implies for b2 being R_eal holds
not ( b2 in REAL & not b1 + b2 <> 0 ) )
proof end;

theorem Th22: :: EXTREAL2:22
for b1 being R_eal holds
( ( b1 = +infty or b1 = -infty ) implies for b2 being R_eal holds
b1 * b2 <> 1. )
proof end;

theorem Th23: :: EXTREAL2:23
for b1, b2 being R_eal holds
not ( not ( b1 = +infty & b2 = -infty ) & not ( b1 = -infty & b2 = +infty ) & b1 + b2 < +infty & not ( b1 <> +infty & b2 <> +infty ) )
proof end;

theorem Th24: :: EXTREAL2:24
for b1, b2 being R_eal holds
not ( not ( b1 = +infty & b2 = -infty ) & not ( b1 = -infty & b2 = +infty ) & -infty < b1 + b2 & not ( b1 <> -infty & b2 <> -infty ) )
proof end;

theorem Th25: :: EXTREAL2:25
for b1, b2 being R_eal holds
not ( not ( b1 = +infty & b2 = +infty ) & not ( b1 = -infty & b2 = -infty ) & b1 - b2 < +infty & not ( b1 <> +infty & b2 <> -infty ) )
proof end;

theorem Th26: :: EXTREAL2:26
for b1, b2 being R_eal holds
not ( not ( b1 = +infty & b2 = +infty ) & not ( b1 = -infty & b2 = -infty ) & -infty < b1 - b2 & not ( b1 <> -infty & b2 <> +infty ) )
proof end;

theorem Th27: :: EXTREAL2:27
for b1, b2, b3 being R_eal holds
not ( not ( b1 = +infty & b2 = -infty ) & not ( b1 = -infty & b2 = +infty ) & b1 + b2 < b3 & not ( b1 <> +infty & b2 <> +infty & b3 <> -infty & b1 < b3 - b2 ) )
proof end;

theorem Th28: :: EXTREAL2:28
for b1, b2, b3 being R_eal holds
not ( not ( b1 = +infty & b2 = +infty ) & not ( b1 = -infty & b2 = -infty ) & b3 < b1 - b2 & not ( b3 <> +infty & b2 <> +infty & b1 <> -infty & b3 + b2 < b1 ) )
proof end;

theorem Th29: :: EXTREAL2:29
for b1, b2, b3 being R_eal holds
not ( not ( b1 = +infty & b2 = +infty ) & not ( b1 = -infty & b2 = -infty ) & b1 - b2 < b3 & not ( b1 <> +infty & b2 <> -infty & b3 <> -infty & b1 < b3 + b2 ) )
proof end;

theorem Th30: :: EXTREAL2:30
for b1, b2, b3 being R_eal holds
not ( not ( b1 = +infty & b2 = -infty ) & not ( b1 = -infty & b2 = +infty ) & b3 < b1 + b2 & not ( b3 <> +infty & b2 <> -infty & b1 <> -infty & b3 - b2 < b1 ) )
proof end;

theorem Th31: :: EXTREAL2:31
for b1, b2, b3 being R_eal holds
not ( not ( b1 = +infty & b2 = -infty ) & not ( b1 = -infty & b2 = +infty ) & not ( b2 = +infty & b3 = +infty ) & not ( b2 = -infty & b3 = -infty ) & b1 + b2 <= b3 & not ( b2 <> +infty & b1 <= b3 - b2 ) )
proof end;

theorem Th32: :: EXTREAL2:32
for b1, b2, b3 being R_eal holds
( not ( b1 = +infty & b2 = -infty ) & not ( b1 = -infty & b2 = +infty ) & not ( b2 = +infty & b3 = +infty ) & b1 <= b3 - b2 implies ( b2 <> +infty & b1 + b2 <= b3 ) )
proof end;

theorem Th33: :: EXTREAL2:33
for b1, b2, b3 being R_eal holds
not ( not ( b1 = +infty & b2 = +infty ) & not ( b1 = -infty & b2 = -infty ) & not ( b2 = +infty & b3 = -infty ) & not ( b2 = -infty & b3 = +infty ) & b1 - b2 <= b3 & not ( b2 <> -infty & b1 <= b3 + b2 ) )
proof end;

theorem Th34: :: EXTREAL2:34
for b1, b2, b3 being R_eal holds
( not ( b1 = +infty & b2 = +infty ) & not ( b1 = -infty & b2 = -infty ) & not ( b2 = -infty & b3 = +infty ) & b1 <= b3 + b2 implies ( b2 <> -infty & b1 - b2 <= b3 ) )
proof end;

theorem Th35: :: EXTREAL2:35
canceled;

theorem Th36: :: EXTREAL2:36
canceled;

theorem Th37: :: EXTREAL2:37
canceled;

theorem Th38: :: EXTREAL2:38
canceled;

theorem Th39: :: EXTREAL2:39
canceled;

theorem Th40: :: EXTREAL2:40
for b1, b2, b3 being R_eal holds
( not ( b1 = +infty & b2 = +infty ) & not ( b1 = -infty & b2 = -infty ) & not ( b2 = +infty & b3 = -infty ) & not ( b2 = -infty & b3 = +infty ) & not ( b1 = +infty & b3 = +infty ) & not ( b1 = -infty & b3 = -infty ) implies (b1 - b2) - b3 = b1 - (b2 + b3) )
proof end;

theorem Th41: :: EXTREAL2:41
for b1, b2, b3 being R_eal holds
( not ( b1 = +infty & b2 = +infty ) & not ( b1 = -infty & b2 = -infty ) & not ( b2 = +infty & b3 = +infty ) & not ( b2 = -infty & b3 = -infty ) & not ( b1 = +infty & b3 = -infty ) & not ( b1 = -infty & b3 = +infty ) implies (b1 - b2) + b3 = b1 - (b2 - b3) )
proof end;

theorem Th42: :: EXTREAL2:42
for b1, b2 being R_eal holds
not ( b1 * b2 <> +infty & b1 * b2 <> -infty & not b1 is Real & not b2 is Real )
proof end;

theorem Th43: :: EXTREAL2:43
for b1, b2 being R_eal holds
( not ( ( ( 0 < b1 & 0 < b2 ) or ( b1 < 0 & b2 < 0 ) ) & not 0 < b1 * b2 ) & not ( 0 < b1 * b2 & not ( 0 < b1 & 0 < b2 ) & not ( b1 < 0 & b2 < 0 ) ) )
proof end;

theorem Th44: :: EXTREAL2:44
for b1, b2 being R_eal holds
( not ( ( ( 0 < b1 & b2 < 0 ) or ( b1 < 0 & 0 < b2 ) ) & not b1 * b2 < 0 ) & not ( b1 * b2 < 0 & not ( 0 < b1 & b2 < 0 ) & not ( b1 < 0 & 0 < b2 ) ) )
proof end;

theorem Th45: :: EXTREAL2:45
for b1, b2 being R_eal holds
( ( ( not ( not 0 <= b1 & not 0 < b1 ) & not ( not 0 <= b2 & not 0 < b2 ) ) or ( not ( not b1 <= 0 & not b1 < 0 ) & not ( not b2 <= 0 & not b2 < 0 ) ) ) iff 0 <= b1 * b2 ) by Th44;

theorem Th46: :: EXTREAL2:46
for b1, b2 being R_eal holds
( ( ( not ( not b1 <= 0 & not b1 < 0 ) & not ( not 0 <= b2 & not 0 < b2 ) ) or ( not ( not 0 <= b1 & not 0 < b1 ) & not ( not b2 <= 0 & not b2 < 0 ) ) ) iff b1 * b2 <= 0 ) by Th43;

theorem Th47: :: EXTREAL2:47
for b1, b2 being R_eal holds
( ( b1 <= - b2 implies b2 <= - b1 ) & ( - b1 <= b2 implies - b2 <= b1 ) )
proof end;

theorem Th48: :: EXTREAL2:48
canceled;

theorem Th49: :: EXTREAL2:49
for b1 being R_eal
for b2 being Real holds
( b1 = b2 implies |.b1.| = abs b2 )
proof end;

theorem Th50: :: EXTREAL2:50
for b1 being R_eal holds
( |.b1.| = b1 or |.b1.| = - b1 )
proof end;

theorem Th51: :: EXTREAL2:51
for b1 being R_eal holds 0 <= |.b1.|
proof end;

theorem Th52: :: EXTREAL2:52
for b1 being R_eal holds
not ( b1 <> 0 & not 0 < |.b1.| )
proof end;

theorem Th53: :: EXTREAL2:53
for b1 being R_eal holds
( b1 = 0 iff |.b1.| = 0 ) by Th52, EXTREAL1:def 3;

theorem Th54: :: EXTREAL2:54
for b1 being R_eal holds
not ( |.b1.| = - b1 & b1 <> 0 & not b1 < 0 )
proof end;

theorem Th55: :: EXTREAL2:55
for b1 being R_eal holds
( b1 <= 0 implies |.b1.| = - b1 )
proof end;

theorem Th56: :: EXTREAL2:56
for b1, b2 being R_eal holds |.(b1 * b2).| = |.b1.| * |.b2.|
proof end;

theorem Th57: :: EXTREAL2:57
for b1 being R_eal holds
( - |.b1.| <= b1 & b1 <= |.b1.| )
proof end;

theorem Th58: :: EXTREAL2:58
for b1, b2 being R_eal holds
( |.b1.| < b2 implies ( - b2 < b1 & b1 < b2 ) )
proof end;

theorem Th59: :: EXTREAL2:59
for b1, b2 being R_eal holds
( - b1 < b2 & b2 < b1 implies ( 0 < b1 & |.b2.| < b1 ) )
proof end;

theorem Th60: :: EXTREAL2:60
for b1, b2 being R_eal holds
( ( - b1 <= b2 & b2 <= b1 ) iff |.b2.| <= b1 )
proof end;

theorem Th61: :: EXTREAL2:61
for b1, b2 being R_eal holds
not ( not ( b1 = +infty & b2 = -infty ) & not ( b1 = -infty & b2 = +infty ) & not |.(b1 + b2).| <= |.b1.| + |.b2.| )
proof end;

theorem Th62: :: EXTREAL2:62
for b1 being R_eal holds
( -infty < b1 & b1 < +infty & b1 <> 0 implies |.b1.| * |.(1. / b1).| = 1. )
proof end;

theorem Th63: :: EXTREAL2:63
for b1 being R_eal holds
( ( b1 = +infty or b1 = -infty ) implies |.b1.| * |.(1. / b1).| = 0 )
proof end;

theorem Th64: :: EXTREAL2:64
for b1 being R_eal holds
( b1 <> 0 implies |.(1. / b1).| = 1. / |.b1.| )
proof end;

theorem Th65: :: EXTREAL2:65
for b1, b2 being R_eal holds
( not ( ( b1 = -infty or b1 = +infty ) & ( b2 = -infty or b2 = +infty ) ) & b2 <> 0 implies |.(b1 / b2).| = |.b1.| / |.b2.| )
proof end;

theorem Th66: :: EXTREAL2:66
for b1 being R_eal holds |.b1.| = |.(- b1).|
proof end;

theorem Th67: :: EXTREAL2:67
for b1 being R_eal holds
( ( b1 = +infty or b1 = -infty ) implies |.b1.| = +infty )
proof end;

theorem Th68: :: EXTREAL2:68
for b1, b2 being R_eal holds
( ( b1 is Real or b2 is Real ) implies |.b1.| - |.b2.| <= |.(b1 - b2).| )
proof end;

theorem Th69: :: EXTREAL2:69
for b1, b2 being R_eal holds
not ( not ( b1 = +infty & b2 = +infty ) & not ( b1 = -infty & b2 = -infty ) & not |.(b1 - b2).| <= |.b1.| + |.b2.| )
proof end;

theorem Th70: :: EXTREAL2:70
for b1 being R_eal holds |.|.b1.|.| = |.b1.|
proof end;

theorem Th71: :: EXTREAL2:71
for b1, b2, b3, b4 being R_eal holds
not ( not ( b1 = +infty & b2 = -infty ) & not ( b1 = -infty & b2 = +infty ) & |.b1.| <= b3 & |.b2.| <= b4 & not |.(b1 + b2).| <= b3 + b4 )
proof end;

theorem Th72: :: EXTREAL2:72
for b1, b2 being R_eal holds
( ( b1 is Real or b2 is Real ) implies |.(|.b1.| - |.b2.|).| <= |.(b1 - b2).| )
proof end;

theorem Th73: :: EXTREAL2:73
for b1, b2 being R_eal holds
( 0 <= b1 * b2 implies |.(b1 + b2).| = |.b1.| + |.b2.| )
proof end;

theorem Th74: :: EXTREAL2:74
for b1, b2 being R_eal
for b3, b4 being Real holds
( b1 = b3 & b2 = b4 implies ( not ( b4 < b3 & not b2 < b1 ) & not ( b2 < b1 & not b4 < b3 ) & ( b4 <= b3 implies b2 <= b1 ) & ( b2 <= b1 implies b4 <= b3 ) ) ) ;

theorem Th75: :: EXTREAL2:75
for b1, b2 being R_eal holds
( ( b1 = -infty or b2 = -infty ) implies min b1,b2 = -infty ) by XXREAL_0:44;

theorem Th76: :: EXTREAL2:76
for b1, b2 being R_eal holds
( ( b1 = +infty or b2 = +infty ) implies max b1,b2 = +infty ) by XXREAL_0:41;

theorem Th77: :: EXTREAL2:77
for b1, b2 being R_eal
for b3, b4 being Real holds
( b1 = b3 & b2 = b4 implies ( min b1,b2 = min b3,b4 & max b1,b2 = max b3,b4 ) ) ;

theorem Th78: :: EXTREAL2:78
for b1, b2 being R_eal holds
( b1 <= b2 implies min b2,b1 = b1 ) by XXREAL_0:def 8;

theorem Th79: :: EXTREAL2:79
for b1, b2 being R_eal holds
( not b1 <= b2 implies min b2,b1 = b2 ) by XXREAL_0:def 8;

theorem Th80: :: EXTREAL2:80
for b1, b2 being R_eal holds
( b1 <> +infty & b2 <> +infty & not ( b1 = +infty & b2 = +infty ) & not ( b1 = -infty & b2 = -infty ) implies min b1,b2 = ((b1 + b2) - |.(b1 - b2).|) / (R_EAL 2) )
proof end;

theorem Th81: :: EXTREAL2:81
for b1, b2 being R_eal holds
( min b1,b2 <= b1 & min b2,b1 <= b1 ) by XXREAL_0:17;

theorem Th82: :: EXTREAL2:82
canceled;

theorem Th83: :: EXTREAL2:83
for b1, b2 being R_eal holds min b1,b2 = min b2,b1 ;

theorem Th84: :: EXTREAL2:84
for b1, b2 being R_eal holds
( min b1,b2 = b1 or min b1,b2 = b2 ) by XXREAL_0:15;

theorem Th85: :: EXTREAL2:85
for b1, b2, b3 being R_eal holds
( ( b1 <= b2 & b1 <= b3 ) iff b1 <= min b2,b3 ) by XXREAL_0:20, XXREAL_0:22;

theorem Th86: :: EXTREAL2:86
canceled;

theorem Th87: :: EXTREAL2:87
for b1, b2 being R_eal holds
( min b1,b2 = b2 implies b2 <= b1 ) by XXREAL_0:def 8;

theorem Th88: :: EXTREAL2:88
for b1, b2, b3 being R_eal holds min b1,(min b2,b3) = min (min b1,b2),b3 by XXREAL_0:33;

theorem Th89: :: EXTREAL2:89
for b1, b2 being R_eal holds
( b1 <= b2 implies max b1,b2 = b2 ) by XXREAL_0:def 9;

theorem Th90: :: EXTREAL2:90
for b1, b2 being R_eal holds
( not b1 <= b2 implies max b1,b2 = b1 ) by XXREAL_0:def 9;

theorem Th91: :: EXTREAL2:91
for b1, b2 being R_eal holds
( b1 <> -infty & b2 <> -infty & not ( b1 = +infty & b2 = +infty ) & not ( b1 = -infty & b2 = -infty ) implies max b1,b2 = ((b1 + b2) + |.(b1 - b2).|) / (R_EAL 2) )
proof end;

theorem Th92: :: EXTREAL2:92
for b1, b2 being R_eal holds
( b1 <= max b1,b2 & b1 <= max b2,b1 )
proof end;

theorem Th93: :: EXTREAL2:93
canceled;

theorem Th94: :: EXTREAL2:94
for b1, b2 being R_eal holds max b1,b2 = max b2,b1 ;

theorem Th95: :: EXTREAL2:95
for b1, b2 being R_eal holds
( max b1,b2 = b1 or max b1,b2 = b2 )
proof end;

theorem Th96: :: EXTREAL2:96
for b1, b2, b3 being R_eal holds
( ( b1 <= b2 & b3 <= b2 ) iff max b1,b3 <= b2 )
proof end;

theorem Th97: :: EXTREAL2:97
canceled;

theorem Th98: :: EXTREAL2:98
for b1, b2 being R_eal holds
( max b1,b2 = b2 implies b1 <= b2 ) by XXREAL_0:def 9;

theorem Th99: :: EXTREAL2:99
for b1, b2, b3 being R_eal holds max b1,(max b2,b3) = max (max b1,b2),b3 by XXREAL_0:34;

definition
let c1, c2 be R_eal;
redefine func max as max c1,c2 -> R_eal;
coherence
max c1,c2 is R_eal
by XXREAL_0:def 1;
redefine func min as min c1,c2 -> R_eal;
coherence
min c1,c2 is R_eal
by XXREAL_0:def 1;
end;

theorem Th100: :: EXTREAL2:100
for b1, b2 being R_eal holds (min b1,b2) + (max b1,b2) = b1 + b2
proof end;

theorem Th101: :: EXTREAL2:101
for b1, b2 being R_eal holds
( max b1,(min b1,b2) = b1 & max (min b1,b2),b1 = b1 & max (min b2,b1),b1 = b1 & max b1,(min b2,b1) = b1 ) by XXREAL_0:36;

theorem Th102: :: EXTREAL2:102
for b1, b2 being R_eal holds
( min b1,(max b1,b2) = b1 & min (max b1,b2),b1 = b1 & min (max b2,b1),b1 = b1 & min b1,(max b2,b1) = b1 ) by XXREAL_0:35;

theorem Th103: :: EXTREAL2:103
for b1, b2, b3 being R_eal holds
( min b1,(max b2,b3) = max (min b1,b2),(min b1,b3) & min (max b2,b3),b1 = max (min b2,b1),(min b3,b1) ) by XXREAL_0:38;

theorem Th104: :: EXTREAL2:104
for b1, b2, b3 being R_eal holds
( max b1,(min b2,b3) = min (max b1,b2),(max b1,b3) & max (min b2,b3),b1 = min (max b2,b1),(max b3,b1) ) by XXREAL_0:39;