:: SUPINF_1 semantic presentation

theorem Th1: :: SUPINF_1:1
not +infty in REAL ;

theorem Th2: :: SUPINF_1:2
canceled;

definition
let IT be set ;
canceled;
attr a1 is +Inf-like means :Def2: :: SUPINF_1:def 2
IT = +infty ;
end;

:: deftheorem Def1 SUPINF_1:def 1 :
canceled;

:: deftheorem Def2 defines +Inf-like SUPINF_1:def 2 :
for IT being set holds
( IT is +Inf-like iff IT = +infty );

registration
cluster +Inf-like set ;
existence
ex b1 being set st b1 is +Inf-like
proof end;
end;

definition
mode +Inf is +Inf-like set ;
end;

theorem Th3: :: SUPINF_1:3
canceled;

theorem Th4: :: SUPINF_1:4
+infty is +Inf by ;

theorem Th5: :: SUPINF_1:5
canceled;

theorem Th6: :: SUPINF_1:6
not -infty in REAL by XREAL_0:def 1;

definition
let IT be set ;
canceled;
attr a1 is -Inf-like means :Def4: :: SUPINF_1:def 4
IT = -infty ;
end;

:: deftheorem Def3 SUPINF_1:def 3 :
canceled;

:: deftheorem Def4 defines -Inf-like SUPINF_1:def 4 :
for IT being set holds
( IT is -Inf-like iff IT = -infty );

registration
cluster -Inf-like set ;
existence
ex b1 being set st b1 is -Inf-like
proof end;
end;

definition
mode -Inf is -Inf-like set ;
end;

theorem Th7: :: SUPINF_1:7
canceled;

theorem Th8: :: SUPINF_1:8
-infty is -Inf by ;

registration
cluster -> ext-real Element of ExtREAL ;
coherence
for b1 being Element of ExtREAL holds b1 is ext-real
;
end;

definition
mode R_eal is Element of ExtREAL ;
end;

registration
cluster -> ext-real Element of REAL ;
coherence
for b1 being Real holds b1 is ext-real
;
end;

theorem Th9: :: SUPINF_1:9
canceled;

theorem Th10: :: SUPINF_1:10
for x being Real holds x is R_eal by XXREAL_0:def 1;

theorem Th11: :: SUPINF_1:11
( -infty is R_eal & +infty is R_eal )
proof end;

definition
redefine func +infty as +infty -> R_eal;
coherence
+infty is R_eal
by ;
redefine func -infty as -infty -> R_eal;
coherence
-infty is R_eal
by ;
end;

theorem Th12: :: SUPINF_1:12
canceled;

theorem Th13: :: SUPINF_1:13
canceled;

theorem Th14: :: SUPINF_1:14
canceled;

definition
let x be R_eal, y be R_eal;
canceled;
canceled;
redefine pred c1 <= c2 means :: SUPINF_1:def 7
ex p, q being Real st
( p = x & q = y & p <= q ) if ( x in REAL & y in REAL )
otherwise ( x = -infty or y = +infty );
consistency
verum
;
compatibility
( ( x in REAL & y in REAL implies ( x <= y iff ex p, q being Real st
( p = x & q = y & p <= q ) ) ) & ( ( not x in REAL or not y in REAL ) implies ( x <= y iff ( x = -infty or y = +infty ) ) ) )
proof end;
end;

:: deftheorem Def5 SUPINF_1:def 5 :
canceled;

:: deftheorem Def6 SUPINF_1:def 6 :
canceled;

:: deftheorem Def7 defines <= SUPINF_1:def 7 :
for x, y being R_eal holds
( ( x in REAL & y in REAL implies ( x <= y iff ex p, q being Real st
( p = x & q = y & p <= q ) ) ) & ( ( not x in REAL or not y in REAL ) implies ( x <= y iff ( x = -infty or y = +infty ) ) ) );

theorem Th15: :: SUPINF_1:15
canceled;

theorem Th16: :: SUPINF_1:16
for x, y being R_eal st x is Real & y is Real holds
( x <= y iff ex p, q being Real st
( p = x & q = y & p <= q ) ) ;

theorem Th17: :: SUPINF_1:17
for x being R_eal st x in REAL holds
-infty < x by XXREAL_0:12;

theorem Th18: :: SUPINF_1:18
for x being R_eal st x in REAL holds
x < +infty by XXREAL_0:9;

theorem Th19: :: SUPINF_1:19
canceled;

theorem Th20: :: SUPINF_1:20
for x being R_eal holds x <= +infty by XXREAL_0:4;

theorem Th21: :: SUPINF_1:21
for x being R_eal holds -infty <= x by XXREAL_0:5;

theorem Th22: :: SUPINF_1:22
for x, y being R_eal st x <= y & y <= x holds
x = y by XXREAL_0:1;

theorem Th23: :: SUPINF_1:23
for x being R_eal st x <= -infty holds
x = -infty by XXREAL_0:6;

theorem Th24: :: SUPINF_1:24
for x being R_eal st +infty <= x holds
x = +infty by XXREAL_0:4;

scheme :: SUPINF_1:sch 31
s31{ P1[ R_eal] } :
ex X being Subset of ExtREAL st
for x being R_eal holds
( x in X iff P1[x] )
proof end;

theorem Th25: :: SUPINF_1:25
canceled;

theorem Th26: :: SUPINF_1:26
canceled;

theorem Th27: :: SUPINF_1:27
for x being set holds
( x is R_eal iff x in ExtREAL ) ;

theorem Th28: :: SUPINF_1:28
canceled;

theorem Th29: :: SUPINF_1:29
for x, y, z being R_eal st x <= y & y <= z holds
x <= z by XXREAL_0:2;

theorem Th30: :: SUPINF_1:30
canceled;

theorem Th31: :: SUPINF_1:31
for x being R_eal st x in REAL holds
( -infty < x & x < +infty )
proof end;

definition
let X be Subset of ExtREAL ;
canceled;
mode majorant of c1 -> R_eal means :Def9: :: SUPINF_1:def 9
for x being R_eal st x in X holds
x <= it;
existence
ex b1 being R_eal st
for x being R_eal st x in X holds
x <= b1
proof end;
end;

:: deftheorem Def8 SUPINF_1:def 8 :
canceled;

:: deftheorem Def9 defines majorant SUPINF_1:def 9 :
for X being Subset of ExtREAL
for b2 being R_eal holds
( b2 is majorant of X iff for x being R_eal st x in X holds
x <= b2 );

theorem Th32: :: SUPINF_1:32
canceled;

theorem Th33: :: SUPINF_1:33
for X being Subset of ExtREAL holds +infty is majorant of X
proof end;

theorem Th34: :: SUPINF_1:34
for X, Y being Subset of ExtREAL st X c= Y holds
for UB being R_eal st UB is majorant of Y holds
UB is majorant of X
proof end;

definition
let X be Subset of ExtREAL ;
mode minorant of c1 -> R_eal means :Def10: :: SUPINF_1:def 10
for x being R_eal st x in X holds
it <= x;
existence
ex b1 being R_eal st
for x being R_eal st x in X holds
b1 <= x
proof end;
end;

:: deftheorem Def10 defines minorant SUPINF_1:def 10 :
for X being Subset of ExtREAL
for b2 being R_eal holds
( b2 is minorant of X iff for x being R_eal st x in X holds
b2 <= x );

theorem Th35: :: SUPINF_1:35
canceled;

theorem Th36: :: SUPINF_1:36
for X being Subset of ExtREAL holds -infty is minorant of X
proof end;

theorem Th37: :: SUPINF_1:37
canceled;

theorem Th38: :: SUPINF_1:38
canceled;

theorem Th39: :: SUPINF_1:39
for X, Y being Subset of ExtREAL st X c= Y holds
for LB being R_eal st LB is minorant of Y holds
LB is minorant of X
proof end;

definition
redefine func REAL as REAL -> non empty Subset of ExtREAL ;
coherence
REAL is non empty Subset of ExtREAL
by XBOOLE_1:7;
end;

theorem Th40: :: SUPINF_1:40
canceled;

theorem Th41: :: SUPINF_1:41
+infty is majorant of REAL by ;

theorem Th42: :: SUPINF_1:42
-infty is minorant of REAL by ;

definition
let X be Subset of ExtREAL ;
attr a1 is bounded_above means :Def11: :: SUPINF_1:def 11
ex UB being majorant of X st UB in REAL ;
end;

:: deftheorem Def11 defines bounded_above SUPINF_1:def 11 :
for X being Subset of ExtREAL holds
( X is bounded_above iff ex UB being majorant of X st UB in REAL );

theorem Th43: :: SUPINF_1:43
canceled;

theorem Th44: :: SUPINF_1:44
for X, Y being Subset of ExtREAL st X c= Y & Y is bounded_above holds
X is bounded_above
proof end;

theorem Th45: :: SUPINF_1:45
not REAL is bounded_above
proof end;

definition
let X be Subset of ExtREAL ;
attr a1 is bounded_below means :Def12: :: SUPINF_1:def 12
ex LB being minorant of X st LB in REAL ;
end;

:: deftheorem Def12 defines bounded_below SUPINF_1:def 12 :
for X being Subset of ExtREAL holds
( X is bounded_below iff ex LB being minorant of X st LB in REAL );

theorem Th46: :: SUPINF_1:46
canceled;

theorem Th47: :: SUPINF_1:47
for X, Y being Subset of ExtREAL st X c= Y & Y is bounded_below holds
X is bounded_below
proof end;

theorem Th48: :: SUPINF_1:48
not REAL is bounded_below
proof end;

definition
let X be Subset of ExtREAL ;
attr a1 is bounded means :Def13: :: SUPINF_1:def 13
( X is bounded_above & X is bounded_below );
end;

:: deftheorem Def13 defines bounded SUPINF_1:def 13 :
for X being Subset of ExtREAL holds
( X is bounded iff ( X is bounded_above & X is bounded_below ) );

theorem Th49: :: SUPINF_1:49
canceled;

theorem Th50: :: SUPINF_1:50
for X, Y being Subset of ExtREAL st X c= Y & Y is bounded holds
X is bounded
proof end;

theorem Th51: :: SUPINF_1:51
for X being Subset of ExtREAL ex Y being non empty Subset of ExtREAL st
for x being R_eal holds
( x in Y iff x is majorant of X )
proof end;

definition
let X be Subset of ExtREAL ;
func SetMajorant c1 -> Subset of ExtREAL means :Def14: :: SUPINF_1:def 14
for x being R_eal holds
( x in it iff x is majorant of X );
existence
ex b1 being Subset of ExtREAL st
for x being R_eal holds
( x in b1 iff x is majorant of X )
proof end;
uniqueness
for b1, b2 being Subset of ExtREAL st ( for x being R_eal holds
( x in b1 iff x is majorant of X ) ) & ( for x being R_eal holds
( x in b2 iff x is majorant of X ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def14 defines SetMajorant SUPINF_1:def 14 :
for X, b2 being Subset of ExtREAL holds
( b2 = SetMajorant X iff for x being R_eal holds
( x in b2 iff x is majorant of X ) );

registration
let X be Subset of ExtREAL ;
cluster SetMajorant a1 -> non empty ;
coherence
not SetMajorant X is empty
proof end;
end;

theorem Th52: :: SUPINF_1:52
canceled;

theorem Th53: :: SUPINF_1:53
canceled;

theorem Th54: :: SUPINF_1:54
for X, Y being Subset of ExtREAL st X c= Y holds
for x being R_eal st x in SetMajorant Y holds
x in SetMajorant X
proof end;

theorem Th55: :: SUPINF_1:55
for X being Subset of ExtREAL ex Y being non empty Subset of ExtREAL st
for x being R_eal holds
( x in Y iff x is minorant of X )
proof end;

definition
let X be Subset of ExtREAL ;
func SetMinorant c1 -> Subset of ExtREAL means :Def15: :: SUPINF_1:def 15
for x being R_eal holds
( x in it iff x is minorant of X );
existence
ex b1 being Subset of ExtREAL st
for x being R_eal holds
( x in b1 iff x is minorant of X )
proof end;
uniqueness
for b1, b2 being Subset of ExtREAL st ( for x being R_eal holds
( x in b1 iff x is minorant of X ) ) & ( for x being R_eal holds
( x in b2 iff x is minorant of X ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def15 defines SetMinorant SUPINF_1:def 15 :
for X, b2 being Subset of ExtREAL holds
( b2 = SetMinorant X iff for x being R_eal holds
( x in b2 iff x is minorant of X ) );

registration
let X be Subset of ExtREAL ;
cluster SetMinorant a1 -> non empty ;
coherence
not SetMinorant X is empty
proof end;
end;

theorem Th56: :: SUPINF_1:56
canceled;

theorem Th57: :: SUPINF_1:57
canceled;

theorem Th58: :: SUPINF_1:58
for X, Y being Subset of ExtREAL st X c= Y holds
for x being R_eal st x in SetMinorant Y holds
x in SetMinorant X
proof end;

theorem Th59: :: SUPINF_1:59
for X being non empty Subset of ExtREAL st X is bounded_above & X <> {-infty } holds
ex x being Real st
( x in X & x <> -infty )
proof end;

theorem Th60: :: SUPINF_1:60
for X being non empty Subset of ExtREAL st X is bounded_below & not X = {+infty } holds
ex x being Real st
( x in X & x <> +infty )
proof end;

theorem Th61: :: SUPINF_1:61
canceled;

theorem Th62: :: SUPINF_1:62
for X being non empty Subset of ExtREAL st X is bounded_above & X <> {-infty } holds
ex UB being R_eal st
( UB is majorant of X & ( for y being R_eal st y is majorant of X holds
UB <= y ) )
proof end;

theorem Th63: :: SUPINF_1:63
for X being non empty Subset of ExtREAL st X is bounded_below & X <> {+infty } holds
ex LB being R_eal st
( LB is minorant of X & ( for y being R_eal st y is minorant of X holds
y <= LB ) )
proof end;

theorem Th64: :: SUPINF_1:64
for X being Subset of ExtREAL st X = {-infty } holds
X is bounded_above
proof end;

theorem Th65: :: SUPINF_1:65
for X being Subset of ExtREAL st X = {+infty } holds
X is bounded_below
proof end;

theorem Th66: :: SUPINF_1:66
for X being Subset of ExtREAL st X = {-infty } holds
ex UB being R_eal st
( UB is majorant of X & ( for y being R_eal st y is majorant of X holds
UB <= y ) )
proof end;

theorem Th67: :: SUPINF_1:67
for X being Subset of ExtREAL st X = {+infty } holds
ex LB being R_eal st
( LB is minorant of X & ( for y being R_eal st y is minorant of X holds
y <= LB ) )
proof end;

theorem Th68: :: SUPINF_1:68
for X being non empty Subset of ExtREAL st X is bounded_above holds
ex UB being R_eal st
( UB is majorant of X & ( for y being R_eal st y is majorant of X holds
UB <= y ) )
proof end;

theorem Th69: :: SUPINF_1:69
for X being non empty Subset of ExtREAL st X is bounded_below holds
ex LB being R_eal st
( LB is minorant of X & ( for y being R_eal st y is minorant of X holds
y <= LB ) )
proof end;

theorem Th70: :: SUPINF_1:70
for X being non empty Subset of ExtREAL st not X is bounded_above holds
for y being R_eal st y is majorant of X holds
y = +infty
proof end;

theorem Th71: :: SUPINF_1:71
for X being non empty Subset of ExtREAL st not X is bounded_below holds
for y being R_eal st y is minorant of X holds
y = -infty
proof end;

theorem Th72: :: SUPINF_1:72
for X being non empty Subset of ExtREAL ex UB being R_eal st
( UB is majorant of X & ( for y being R_eal st y is majorant of X holds
UB <= y ) )
proof end;

theorem Th73: :: SUPINF_1:73
for X being non empty Subset of ExtREAL ex LB being R_eal st
( LB is minorant of X & ( for y being R_eal st y is minorant of X holds
y <= LB ) )
proof end;

definition
let X be non empty Subset of ExtREAL ;
func sup c1 -> R_eal means :Def16: :: SUPINF_1:def 16
( it is majorant of X & ( for y being R_eal st y is majorant of X holds
it <= y ) );
existence
ex b1 being R_eal st
( b1 is majorant of X & ( for y being R_eal st y is majorant of X holds
b1 <= y ) )
by ;
uniqueness
for b1, b2 being R_eal st b1 is majorant of X & ( for y being R_eal st y is majorant of X holds
b1 <= y ) & b2 is majorant of X & ( for y being R_eal st y is majorant of X holds
b2 <= y ) holds
b1 = b2
proof end;
end;

:: deftheorem Def16 defines sup SUPINF_1:def 16 :
for X being non empty Subset of ExtREAL
for b2 being R_eal holds
( b2 = sup X iff ( b2 is majorant of X & ( for y being R_eal st y is majorant of X holds
b2 <= y ) ) );

theorem Th74: :: SUPINF_1:74
canceled;

theorem Th75: :: SUPINF_1:75
canceled;

theorem Th76: :: SUPINF_1:76
for X being non empty Subset of ExtREAL
for x being R_eal st x in X holds
x <= sup X
proof end;

definition
let X be non empty Subset of ExtREAL ;
func inf c1 -> R_eal means :Def17: :: SUPINF_1:def 17
( it is minorant of X & ( for y being R_eal st y is minorant of X holds
y <= it ) );
existence
ex b1 being R_eal st
( b1 is minorant of X & ( for y being R_eal st y is minorant of X holds
y <= b1 ) )
by Th29;
uniqueness
for b1, b2 being R_eal st b1 is minorant of X & ( for y being R_eal st y is minorant of X holds
y <= b1 ) & b2 is minorant of X & ( for y being R_eal st y is minorant of X holds
y <= b2 ) holds
b1 = b2
proof end;
end;

:: deftheorem Def17 defines inf SUPINF_1:def 17 :
for X being non empty Subset of ExtREAL
for b2 being R_eal holds
( b2 = inf X iff ( b2 is minorant of X & ( for y being R_eal st y is minorant of X holds
y <= b2 ) ) );

theorem Th77: :: SUPINF_1:77
canceled;

theorem Th78: :: SUPINF_1:78
canceled;

theorem Th79: :: SUPINF_1:79
for X being non empty Subset of ExtREAL
for x being R_eal st x in X holds
inf X <= x
proof end;

theorem Th80: :: SUPINF_1:80
for X being non empty Subset of ExtREAL
for x being majorant of X st x in X holds
x = sup X
proof end;

theorem Th81: :: SUPINF_1:81
for X being non empty Subset of ExtREAL
for x being minorant of X st x in X holds
x = inf X
proof end;

theorem Th82: :: SUPINF_1:82
for X being non empty Subset of ExtREAL holds
( sup X = inf (SetMajorant X) & inf X = sup (SetMinorant X) )
proof end;

theorem Th83: :: SUPINF_1:83
for X being non empty Subset of ExtREAL st X is bounded_above & X <> {-infty } holds
sup X in REAL
proof end;

theorem Th84: :: SUPINF_1:84
for X being non empty Subset of ExtREAL st X is bounded_below & X <> {+infty } holds
inf X in REAL
proof end;

theorem Th85: :: SUPINF_1:85
for x being R_eal holds sup {x} = x
proof end;

theorem Th86: :: SUPINF_1:86
for x being R_eal holds inf {x} = x
proof end;

theorem Th87: :: SUPINF_1:87
canceled;

theorem Th88: :: SUPINF_1:88
canceled;

theorem Th89: :: SUPINF_1:89
canceled;

theorem Th90: :: SUPINF_1:90
canceled;

theorem Th91: :: SUPINF_1:91
for X, Y being non empty Subset of ExtREAL st X c= Y holds
sup X <= sup Y
proof end;

theorem Th92: :: SUPINF_1:92
for x, y, a being R_eal st x <= a & y <= a holds
sup {x,y} <= a
proof end;

theorem Th93: :: SUPINF_1:93
for x, y being R_eal holds
( ( x <= y implies sup {x,y} = y ) & ( y <= x implies sup {x,y} = x ) )
proof end;

theorem Th94: :: SUPINF_1:94
for X, Y being non empty Subset of ExtREAL st X c= Y holds
inf Y <= inf X
proof end;

theorem Th95: :: SUPINF_1:95
for x, y, a being R_eal st a <= x & a <= y holds
a <= inf {x,y}
proof end;

theorem Th96: :: SUPINF_1:96
for x, y being R_eal holds
( ( x <= y implies inf {x,y} = x ) & ( y <= x implies inf {x,y} = y ) )
proof end;

theorem Th97: :: SUPINF_1:97
for X being non empty Subset of ExtREAL
for x being R_eal st ex y being R_eal st
( y in X & x <= y ) holds
x <= sup X
proof end;

theorem Th98: :: SUPINF_1:98
for X being non empty Subset of ExtREAL
for x being R_eal st ex y being R_eal st
( y in X & y <= x ) holds
inf X <= x
proof end;

theorem Th99: :: SUPINF_1:99
for X, Y being non empty Subset of ExtREAL st ( for x being R_eal st x in X holds
ex y being R_eal st
( y in Y & x <= y ) ) holds
sup X <= sup Y
proof end;

theorem Th100: :: SUPINF_1:100
for X, Y being non empty Subset of ExtREAL st ( for y being R_eal st y in Y holds
ex x being R_eal st
( x in X & x <= y ) ) holds
inf X <= inf Y
proof end;

theorem Th101: :: SUPINF_1:101
for X, Y being Subset of ExtREAL
for UB1 being majorant of X
for UB2 being majorant of Y holds sup {UB1,UB2} is majorant of X \/ Y
proof end;

theorem Th102: :: SUPINF_1:102
for X, Y being Subset of ExtREAL
for LB1 being minorant of X
for LB2 being minorant of Y holds inf {LB1,LB2} is minorant of X \/ Y
proof end;

theorem Th103: :: SUPINF_1:103
for X, Y, S being Subset of ExtREAL
for UB1 being majorant of X
for UB2 being majorant of Y st S = X /\ Y holds
inf {UB1,UB2} is majorant of S
proof end;

theorem Th104: :: SUPINF_1:104
for X, Y, S being Subset of ExtREAL
for LB1 being minorant of X
for LB2 being minorant of Y st S = X /\ Y holds
sup {LB1,LB2} is minorant of S
proof end;

theorem Th105: :: SUPINF_1:105
for X, Y being non empty Subset of ExtREAL holds sup (X \/ Y) = sup {(sup X),(sup Y)}
proof end;

theorem Th106: :: SUPINF_1:106
for X, Y being non empty Subset of ExtREAL holds inf (X \/ Y) = inf {(inf X),(inf Y)}
proof end;

theorem Th107: :: SUPINF_1:107
for X, Y, S being non empty Subset of ExtREAL st S = X /\ Y holds
sup S <= inf {(sup X),(sup Y)}
proof end;

theorem Th108: :: SUPINF_1:108
for X, Y, S being non empty Subset of ExtREAL st S = X /\ Y holds
sup {(inf X),(inf Y)} <= inf S
proof end;

definition
let X be non empty set ;
mode bool_DOMAIN of c1 -> set means :Def18: :: SUPINF_1:def 18
( it is non empty Subset-Family of X & ( for A being set st A in it holds
not A is empty ) );
existence
ex b1 being set st
( b1 is non empty Subset-Family of X & ( for A being set st A in b1 holds
not A is empty ) )
proof end;
end;

:: deftheorem Def18 defines bool_DOMAIN SUPINF_1:def 18 :
for X being non empty set
for b2 being set holds
( b2 is bool_DOMAIN of X iff ( b2 is non empty Subset-Family of X & ( for A being set st A in b2 holds
not A is empty ) ) );

definition
let F be bool_DOMAIN of ExtREAL ;
func SUP c1 -> Subset of ExtREAL means :Def19: :: SUPINF_1:def 19
for a being R_eal holds
( a in it iff ex A being non empty Subset of ExtREAL st
( A in F & a = sup A ) );
existence
ex b1 being Subset of ExtREAL st
for a being R_eal holds
( a in b1 iff ex A being non empty Subset of ExtREAL st
( A in F & a = sup A ) )
proof end;
uniqueness
for b1, b2 being Subset of ExtREAL st ( for a being R_eal holds
( a in b1 iff ex A being non empty Subset of ExtREAL st
( A in F & a = sup A ) ) ) & ( for a being R_eal holds
( a in b2 iff ex A being non empty Subset of ExtREAL st
( A in F & a = sup A ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def19 defines SUP SUPINF_1:def 19 :
for F being bool_DOMAIN of ExtREAL
for b2 being Subset of ExtREAL holds
( b2 = SUP F iff for a being R_eal holds
( a in b2 iff ex A being non empty Subset of ExtREAL st
( A in F & a = sup A ) ) );

registration
let F be bool_DOMAIN of ExtREAL ;
cluster SUP a1 -> non empty ;
coherence
not SUP F is empty
proof end;
end;

theorem Th109: :: SUPINF_1:109
canceled;

theorem Th110: :: SUPINF_1:110
canceled;

theorem Th111: :: SUPINF_1:111
canceled;

theorem Th112: :: SUPINF_1:112
for F being bool_DOMAIN of ExtREAL
for S being non empty Subset of ExtREAL st S = union F holds
sup S is majorant of SUP F
proof end;

theorem Th113: :: SUPINF_1:113
for F being bool_DOMAIN of ExtREAL
for S being Subset of ExtREAL st S = union F holds
sup (SUP F) is majorant of S
proof end;

theorem Th114: :: SUPINF_1:114
for F being bool_DOMAIN of ExtREAL
for S being non empty Subset of ExtREAL st S = union F holds
sup S = sup (SUP F)
proof end;

definition
let F be bool_DOMAIN of ExtREAL ;
func INF c1 -> Subset of ExtREAL means :Def20: :: SUPINF_1:def 20
for a being R_eal holds
( a in it iff ex A being non empty Subset of ExtREAL st
( A in F & a = inf A ) );
existence
ex b1 being Subset of ExtREAL st
for a being R_eal holds
( a in b1 iff ex A being non empty Subset of ExtREAL st
( A in F & a = inf A ) )
proof end;
uniqueness
for b1, b2 being Subset of ExtREAL st ( for a being R_eal holds
( a in b1 iff ex A being non empty Subset of ExtREAL st
( A in F & a = inf A ) ) ) & ( for a being R_eal holds
( a in b2 iff ex A being non empty Subset of ExtREAL st
( A in F & a = inf A ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def20 defines INF SUPINF_1:def 20 :
for F being bool_DOMAIN of ExtREAL
for b2 being Subset of ExtREAL holds
( b2 = INF F iff for a being R_eal holds
( a in b2 iff ex A being non empty Subset of ExtREAL st
( A in F & a = inf A ) ) );

registration
let F be bool_DOMAIN of ExtREAL ;
cluster INF a1 -> non empty ;
coherence
not INF F is empty
proof end;
end;

theorem Th115: :: SUPINF_1:115
canceled;

theorem Th116: :: SUPINF_1:116
canceled;

theorem Th117: :: SUPINF_1:117
for F being bool_DOMAIN of ExtREAL
for S being non empty Subset of ExtREAL st S = union F holds
inf S is minorant of INF F
proof end;

theorem Th118: :: SUPINF_1:118
for F being bool_DOMAIN of ExtREAL
for S being Subset of ExtREAL st S = union F holds
inf (INF F) is minorant of S
proof end;

theorem Th119: :: SUPINF_1:119
for F being bool_DOMAIN of ExtREAL
for S being non empty Subset of ExtREAL st S = union F holds
inf S = inf (INF F)
proof end;

theorem Th120: :: SUPINF_1:120
for x, y being R_eal
for p, q being real number st x = p & y = q holds
( p <= q iff x <= y ) ;