:: Series of Positive Real Numbers. Measure Theory
:: by J\'ozef Bia{\l}as
::
:: Received September 27, 1990
:: Copyright (c) 1990-2011 Association of Mizar Users


begin

notation
synonym 0. for 0 ;
end;

definition
:: original: 0.
redefine func 0. -> R_eal;
coherence
0. is R_eal by XXREAL_0:def 1;
let x be R_eal;
:: original: -
redefine func - x -> R_eal;
coherence
- x is R_eal by XXREAL_0:def 1;
let y be R_eal;
:: original: +
redefine func x + y -> R_eal;
coherence
x + y is R_eal by XXREAL_0:def 1;
:: original: -
redefine func x - y -> R_eal;
coherence
x - y is R_eal by XXREAL_0:def 1;
end;

theorem :: SUPINF_2:1
for x, y being R_eal
for a, b being Real st x = a & y = b holds
x + y = a + b by XXREAL_3:def 2;

theorem :: SUPINF_2:2
canceled;

theorem :: SUPINF_2:3
for x being R_eal
for a being Real st x = a holds
- x = - a by XXREAL_3:def 3;

theorem :: SUPINF_2:4
canceled;

theorem :: SUPINF_2:5
for x, y being R_eal
for a, b being Real st x = a & y = b holds
x - y = a - b
proof end;

theorem :: SUPINF_2:6
canceled;

theorem :: SUPINF_2:7
canceled;

theorem :: SUPINF_2:8
canceled;

theorem :: SUPINF_2:9
canceled;

theorem :: SUPINF_2:10
canceled;

theorem :: SUPINF_2:11
canceled;

theorem :: SUPINF_2:12
canceled;

theorem :: SUPINF_2:13
canceled;

theorem :: SUPINF_2:14
canceled;

theorem :: SUPINF_2:15
canceled;

theorem :: SUPINF_2:16
canceled;

theorem :: SUPINF_2:17
canceled;

theorem :: SUPINF_2:18
canceled;

theorem :: SUPINF_2:19
canceled;

theorem :: SUPINF_2:20
canceled;

notation
let X, Y be Subset of ExtREAL;
synonym X + Y for X ++ Y;
end;

definition
canceled;
canceled;
canceled;
canceled;
canceled;
canceled;
let X, Y be Subset of ExtREAL;
:: original: +
redefine func X + Y -> Subset of ExtREAL;
coherence
X + Y is Subset of ExtREAL by MEMBERED:2;
end;

:: deftheorem SUPINF_2:def 1 :
canceled;

:: deftheorem SUPINF_2:def 2 :
canceled;

:: deftheorem SUPINF_2:def 3 :
canceled;

:: deftheorem SUPINF_2:def 4 :
canceled;

:: deftheorem SUPINF_2:def 5 :
canceled;

:: deftheorem SUPINF_2:def 6 :
canceled;

notation
let X be Subset of ExtREAL;
synonym - X for -- X;
end;

definition
let X be Subset of ExtREAL;
:: original: -
redefine func - X -> Subset of ExtREAL;
coherence
- X is Subset of ExtREAL by MEMBERED:2;
end;

theorem :: SUPINF_2:21
canceled;

theorem :: SUPINF_2:22
for X being non empty Subset of ExtREAL holds - (- X) = X ;

theorem :: SUPINF_2:23
for X being non empty Subset of ExtREAL
for z being R_eal holds
( z in X iff - z in - X ) by MEMBER_1:1;

theorem :: SUPINF_2:24
for X, Y being non empty Subset of ExtREAL holds
( X c= Y iff - X c= - Y ) by MEMBER_1:3;

theorem :: SUPINF_2:25
for z being R_eal holds
( z in REAL iff - z in REAL )
proof end;

definition
let X be ext-real-membered set ;
:: original: inf
redefine func inf X -> R_eal;
coherence
inf X is R_eal by XXREAL_0:def 1;
:: original: sup
redefine func sup X -> R_eal;
coherence
sup X is R_eal by XXREAL_0:def 1;
end;

theorem Th26: :: SUPINF_2:26
for X, Y being non empty Subset of ExtREAL holds sup (X + Y) <= (sup X) + (sup Y)
proof end;

theorem Th27: :: SUPINF_2:27
for X, Y being non empty Subset of ExtREAL holds (inf X) + (inf Y) <= inf (X + Y)
proof end;

theorem :: SUPINF_2:28
for X, Y being non empty Subset of ExtREAL st X is bounded_above & Y is bounded_above holds
sup (X + Y) <= (sup X) + (sup Y) by Th26;

theorem :: SUPINF_2:29
for X, Y being non empty Subset of ExtREAL st X is bounded_below & Y is bounded_below holds
(inf X) + (inf Y) <= inf (X + Y) by Th27;

theorem Th30: :: SUPINF_2:30
for X being non empty Subset of ExtREAL
for a being R_eal holds
( a is UpperBound of X iff - a is LowerBound of - X )
proof end;

theorem Th31: :: SUPINF_2:31
for X being non empty Subset of ExtREAL
for a being R_eal holds
( a is LowerBound of X iff - a is UpperBound of - X )
proof end;

theorem Th32: :: SUPINF_2:32
for X being non empty Subset of ExtREAL holds inf (- X) = - (sup X)
proof end;

theorem Th33: :: SUPINF_2:33
for X being non empty Subset of ExtREAL holds sup (- X) = - (inf X)
proof end;

definition
let X be non empty set ;
let Y be non empty Subset of ExtREAL;
let F be Function of X,Y;
:: original: proj2
redefine func rng F -> non empty Subset of ExtREAL;
coherence
proj2 F is non empty Subset of ExtREAL
proof end;
end;

definition
let D be ext-real-membered set ;
let X be set ;
let Y be Subset of D;
let F be PartFunc of X,Y;
func sup F -> Element of ExtREAL equals :: SUPINF_2:def 7
 sup (rng F);
coherence
sup (rng F) is Element of ExtREAL ;
func inf F -> Element of ExtREAL equals :: SUPINF_2:def 8
 inf (rng F);
coherence
inf (rng F) is Element of ExtREAL ;
end;

:: deftheorem defines sup SUPINF_2:def 7 :
for D being ext-real-membered set
for X being set
for Y being Subset of D
for F being PartFunc of X,Y holds sup F = sup (rng F);

:: deftheorem defines inf SUPINF_2:def 8 :
for D being ext-real-membered set
for X being set
for Y being Subset of D
for F being PartFunc of X,Y holds inf F = inf (rng F);

definition
let F be ext-real-valued Function;
let x be set ;
:: original: .
redefine func F . x -> R_eal;
coherence
F . x is R_eal by XXREAL_0:def 1;
end;

definition
let X be non empty set ;
let Y, Z be non empty Subset of ExtREAL;
let F be Function of X,Y;
let G be Function of X,Z;
func F + G -> Function of X,(Y + Z) means :Def9: :: SUPINF_2:def 9
 for x being Element of X holds it . x = (F . x) + (G . x);
existence
ex b1 being Function of X,(Y + Z) st
for x being Element of X holds b1 . x = (F . x) + (G . x)
proof end;
uniqueness
for b1, b2 being Function of X,(Y + Z) st ( for x being Element of X holds b1 . x = (F . x) + (G . x) ) & ( for x being Element of X holds b2 . x = (F . x) + (G . x) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def9 defines + SUPINF_2:def 9 :
for X being non empty set
for Y, Z being non empty Subset of ExtREAL
for F being Function of X,Y
for G being Function of X,Z
for b6 being Function of X,(Y + Z) holds
( b6 = F + G iff for x being Element of X holds b6 . x = (F . x) + (G . x) );

theorem Th34: :: SUPINF_2:34
for X being non empty set
for Y, Z being non empty Subset of ExtREAL
for F being Function of X,Y
for G being Function of X,Z holds rng (F + G) c= (rng F) + (rng G)
proof end;

theorem Th35: :: SUPINF_2:35
for X being non empty set
for Y, Z being non empty Subset of ExtREAL
for F being Function of X,Y
for G being Function of X,Z holds sup (F + G) <= (sup F) + (sup G)
proof end;

theorem Th36: :: SUPINF_2:36
for X being non empty set
for Y, Z being non empty Subset of ExtREAL
for F being Function of X,Y
for G being Function of X,Z holds (inf F) + (inf G) <= inf (F + G)
proof end;

definition
let X be non empty set ;
let Y be non empty Subset of ExtREAL;
let F be Function of X,Y;
func - F -> Function of X,(- Y) means :Def10: :: SUPINF_2:def 10
 for x being Element of X holds it . x = - (F . x);
existence
ex b1 being Function of X,(- Y) st
for x being Element of X holds b1 . x = - (F . x)
proof end;
uniqueness
for b1, b2 being Function of X,(- Y) st ( for x being Element of X holds b1 . x = - (F . x) ) & ( for x being Element of X holds b2 . x = - (F . x) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def10 defines - SUPINF_2:def 10 :
for X being non empty set
for Y being non empty Subset of ExtREAL
for F being Function of X,Y
for b4 being Function of X,(- Y) holds
( b4 = - F iff for x being Element of X holds b4 . x = - (F . x) );

theorem Th37: :: SUPINF_2:37
for X being non empty set
for Y being non empty Subset of ExtREAL
for F being Function of X,Y holds rng (- F) = - (rng F)
proof end;

theorem Th38: :: SUPINF_2:38
for X being non empty set
for Y being non empty Subset of ExtREAL
for F being Function of X,Y holds
( inf (- F) = - (sup F) & sup (- F) = - (inf F) )
proof end;

definition
let X be set ;
let Y be Subset of ExtREAL;
let F be Function of X,Y;
attr F is bounded_above means :Def11: :: SUPINF_2:def 11
 sup F < +infty ;
attr F is bounded_below means :Def12: :: SUPINF_2:def 12
 -infty < inf F;
end;

:: deftheorem Def11 defines bounded_above SUPINF_2:def 11 :
for X being set
for Y being Subset of ExtREAL
for F being Function of X,Y holds
( F is bounded_above iff sup F < +infty );

:: deftheorem Def12 defines bounded_below SUPINF_2:def 12 :
for X being set
for Y being Subset of ExtREAL
for F being Function of X,Y holds
( F is bounded_below iff -infty < inf F );

definition
let X be set ;
let Y be Subset of ExtREAL;
let F be Function of X,Y;
attr F is bounded means :Def13: :: SUPINF_2:def 13
( F is bounded_above & F is bounded_below );
end;

:: deftheorem Def13 defines bounded SUPINF_2:def 13 :
for X being set
for Y being Subset of ExtREAL
for F being Function of X,Y holds
( F is bounded iff ( F is bounded_above & F is bounded_below ) );

registration
let X be set ;
let Y be Subset of ExtREAL;
cluster Function-like V15(X,Y) bounded -> bounded_above bounded_below Element of K21(K22(X,Y));
coherence
for b1 being Function of X,Y st b1 is bounded holds
( b1 is bounded_above & b1 is bounded_below ) by Def13;
cluster Function-like V15(X,Y) bounded_above bounded_below -> bounded Element of K21(K22(X,Y));
coherence
for b1 being Function of X,Y st b1 is bounded_above & b1 is bounded_below holds
b1 is bounded by Def13;
end;

theorem :: SUPINF_2:39
for X being non empty set
for Y being non empty Subset of ExtREAL
for F being Function of X,Y holds
( F is bounded iff ( sup F < +infty & -infty < inf F ) )
proof end;

theorem Th40: :: SUPINF_2:40
for X being non empty set
for Y being non empty Subset of ExtREAL
for F being Function of X,Y holds
( F is bounded_above iff - F is bounded_below )
proof end;

theorem Th41: :: SUPINF_2:41
for X being non empty set
for Y being non empty Subset of ExtREAL
for F being Function of X,Y holds
( F is bounded_below iff - F is bounded_above )
proof end;

theorem :: SUPINF_2:42
for X being non empty set
for Y being non empty Subset of ExtREAL
for F being Function of X,Y holds
( F is bounded iff - F is bounded )
proof end;

theorem :: SUPINF_2:43
for X being non empty set
for Y being non empty Subset of ExtREAL
for F being Function of X,Y
for x being Element of X holds
( -infty <= F . x & F . x <= +infty ) by XXREAL_0:4, XXREAL_0:5;

theorem :: SUPINF_2:44
for X being non empty set
for Y being non empty Subset of ExtREAL
for F being Function of X,Y
for x being Element of X st Y c= REAL holds
( -infty < F . x & F . x < +infty )
proof end;

theorem Th45: :: SUPINF_2:45
for X being non empty set
for Y being non empty Subset of ExtREAL
for F being Function of X,Y
for x being Element of X holds
( inf F <= F . x & F . x <= sup F )
proof end;

theorem Th46: :: SUPINF_2:46
for X being non empty set
for Y being non empty Subset of ExtREAL
for F being Function of X,Y st Y c= REAL holds
( F is bounded_above iff sup F in REAL )
proof end;

theorem Th47: :: SUPINF_2:47
for X being non empty set
for Y being non empty Subset of ExtREAL
for F being Function of X,Y st Y c= REAL holds
( F is bounded_below iff inf F in REAL )
proof end;

theorem :: SUPINF_2:48
for X being non empty set
for Y being non empty Subset of ExtREAL
for F being Function of X,Y st Y c= REAL holds
( F is bounded iff ( inf F in REAL & sup F in REAL ) )
proof end;

theorem Th49: :: SUPINF_2:49
for X being non empty set
for Y, Z being non empty Subset of ExtREAL
for F1 being Function of X,Y
for F2 being Function of X,Z st F1 is bounded_above & F2 is bounded_above holds
F1 + F2 is bounded_above
proof end;

theorem Th50: :: SUPINF_2:50
for X being non empty set
for Y, Z being non empty Subset of ExtREAL
for F1 being Function of X,Y
for F2 being Function of X,Z st F1 is bounded_below & F2 is bounded_below holds
F1 + F2 is bounded_below
proof end;

theorem :: SUPINF_2:51
for X being non empty set
for Y, Z being non empty Subset of ExtREAL
for F1 being Function of X,Y
for F2 being Function of X,Z st F1 is bounded & F2 is bounded holds
F1 + F2 is bounded
proof end;

theorem Th52: :: SUPINF_2:52
( incl NAT is Function of NAT,ExtREAL & incl NAT is one-to-one & NAT = rng (incl NAT) & rng (incl NAT) is non empty Subset of ExtREAL )
proof end;

definition
let D be non empty set ;
let IT be Subset of D;
:: original: countable
redefine attr IT is countable means :Def14: :: SUPINF_2:def 14
( IT is empty or ex F being Function of NAT,D st IT = rng F );
compatibility
( IT is countable iff ( IT is empty or ex F being Function of NAT,D st IT = rng F ) )
proof end;
end;

:: deftheorem Def14 defines countable SUPINF_2:def 14 :
for D being non empty set
for IT being Subset of D holds
( IT is countable iff ( IT is empty or ex F being Function of NAT,D st IT = rng F ) );

registration
cluster non empty ext-real-membered V75() Element of K21(ExtREAL);
existence
ex b1 being non empty Subset of ExtREAL st b1 is countable
proof end;
end;

definition
mode Denum_Set_of_R_EAL is non empty V75() Subset of ExtREAL;
end;

definition
let IT be set ;
attr IT is nonnegative means :Def15: :: SUPINF_2:def 15
 for x being R_eal st x in IT holds
0. <= x;
end;

:: deftheorem Def15 defines nonnegative SUPINF_2:def 15 :
for IT being set holds
( IT is nonnegative iff for x being R_eal st x in IT holds
0. <= x );

registration
cluster non empty ext-real-membered countable nonnegative Element of K21(ExtREAL);
existence
ex b1 being Denum_Set_of_R_EAL st b1 is nonnegative
proof end;
end;

definition
mode Pos_Denum_Set_of_R_EAL is nonnegative Denum_Set_of_R_EAL;
end;

definition
let D be Denum_Set_of_R_EAL;
mode Num of D -> Function of NAT,ExtREAL means :Def16: :: SUPINF_2:def 16
D = rng it;
existence
ex b1 being Function of NAT,ExtREAL st D = rng b1 by Def14;
end;

:: deftheorem Def16 defines Num SUPINF_2:def 16 :
for D being Denum_Set_of_R_EAL
for b2 being Function of NAT,ExtREAL holds
( b2 is Num of D iff D = rng b2 );

theorem Th53: :: SUPINF_2:53
for D being Denum_Set_of_R_EAL
for N being Num of D ex F being Function of NAT,ExtREAL st
( F . 0 = N . 0 & ( for n being Element of NAT
for y being R_eal st y = F . n holds
F . (n + 1) = y + (N . (n + 1)) ) )
proof end;

definition
let D be Denum_Set_of_R_EAL;
let N be Num of D;
func Ser (D,N) -> Function of NAT,ExtREAL means :Def17: :: SUPINF_2:def 17
( it . 0 = N . 0 & ( for n being Element of NAT
for y being R_eal st y = it . n holds
it . (n + 1) = y + (N . (n + 1)) ) );
existence
ex b1 being Function of NAT,ExtREAL st
( b1 . 0 = N . 0 & ( for n being Element of NAT
for y being R_eal st y = b1 . n holds
b1 . (n + 1) = y + (N . (n + 1)) ) ) by Th53;
uniqueness
for b1, b2 being Function of NAT,ExtREAL st b1 . 0 = N . 0 & ( for n being Element of NAT
for y being R_eal st y = b1 . n holds
b1 . (n + 1) = y + (N . (n + 1)) ) & b2 . 0 = N . 0 & ( for n being Element of NAT
for y being R_eal st y = b2 . n holds
b2 . (n + 1) = y + (N . (n + 1)) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def17 defines Ser SUPINF_2:def 17 :
for D being Denum_Set_of_R_EAL
for N being Num of D
for b3 being Function of NAT,ExtREAL holds
( b3 = Ser (D,N) iff ( b3 . 0 = N . 0 & ( for n being Element of NAT
for y being R_eal st y = b3 . n holds
b3 . (n + 1) = y + (N . (n + 1)) ) ) );

theorem Th54: :: SUPINF_2:54
for D being Pos_Denum_Set_of_R_EAL
for N being Num of D
for n being Element of NAT holds 0. <= N . n
proof end;

theorem Th55: :: SUPINF_2:55
for D being Pos_Denum_Set_of_R_EAL
for N being Num of D
for n being Element of NAT holds
( (Ser (D,N)) . n <= (Ser (D,N)) . (n + 1) & 0. <= (Ser (D,N)) . n )
proof end;

theorem Th56: :: SUPINF_2:56
for D being Pos_Denum_Set_of_R_EAL
for N being Num of D
for n, m being Element of NAT holds (Ser (D,N)) . n <= (Ser (D,N)) . (n + m)
proof end;

definition
let D be Denum_Set_of_R_EAL;
mode Set_of_Series of D -> non empty Subset of ExtREAL means :: SUPINF_2:def 18
 ex N being Num of D st it = rng (Ser (D,N));
existence
ex b1 being non empty Subset of ExtREAL ex N being Num of D st b1 = rng (Ser (D,N))
proof end;
end;

:: deftheorem defines Set_of_Series SUPINF_2:def 18 :
for D being Denum_Set_of_R_EAL
for b2 being non empty Subset of ExtREAL holds
( b2 is Set_of_Series of D iff ex N being Num of D st b2 = rng (Ser (D,N)) );

definition
let D be Pos_Denum_Set_of_R_EAL;
let N be Num of D;
func SUM (D,N) -> R_eal equals :: SUPINF_2:def 19
 sup (rng (Ser (D,N)));
coherence
sup (rng (Ser (D,N))) is R_eal ;
end;

:: deftheorem defines SUM SUPINF_2:def 19 :
for D being Pos_Denum_Set_of_R_EAL
for N being Num of D holds SUM (D,N) = sup (rng (Ser (D,N)));

definition
let D be Pos_Denum_Set_of_R_EAL;
let N be Num of D;
pred D is_sumable N means :: SUPINF_2:def 20
 SUM (D,N) in REAL ;
end;

:: deftheorem defines is_sumable SUPINF_2:def 20 :
for D being Pos_Denum_Set_of_R_EAL
for N being Num of D holds
( D is_sumable N iff SUM (D,N) in REAL );

theorem Th57: :: SUPINF_2:57
for F being Function of NAT,ExtREAL holds rng F is Denum_Set_of_R_EAL by Def14;

definition
let F be Function of NAT,ExtREAL;
:: original: proj2
redefine func rng F -> Denum_Set_of_R_EAL;
coherence
proj2 F is Denum_Set_of_R_EAL by Th57;
end;

definition
let F be Function of NAT,ExtREAL;
func Ser F -> Function of NAT,ExtREAL means :Def21: :: SUPINF_2:def 21
 for N being Num of rng F st N = F holds
it = Ser ((rng F),N);
existence
ex b1 being Function of NAT,ExtREAL st
for N being Num of rng F st N = F holds
b1 = Ser ((rng F),N)
proof end;
uniqueness
for b1, b2 being Function of NAT,ExtREAL st ( for N being Num of rng F st N = F holds
b1 = Ser ((rng F),N) ) & ( for N being Num of rng F st N = F holds
b2 = Ser ((rng F),N) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def21 defines Ser SUPINF_2:def 21 :
for F, b2 being Function of NAT,ExtREAL holds
( b2 = Ser F iff for N being Num of rng F st N = F holds
b2 = Ser ((rng F),N) );

definition
let R be Relation;
attr R is nonnegative means :Def22: :: SUPINF_2:def 22
 rng R is nonnegative ;
end;

:: deftheorem Def22 defines nonnegative SUPINF_2:def 22 :
for R being Relation holds
( R is nonnegative iff rng R is nonnegative );

definition
let F be Function of NAT,ExtREAL;
func SUM F -> R_eal equals :: SUPINF_2:def 23
 sup (rng (Ser F));
coherence
sup (rng (Ser F)) is R_eal ;
end;

:: deftheorem defines SUM SUPINF_2:def 23 :
for F being Function of NAT,ExtREAL holds SUM F = sup (rng (Ser F));

theorem Th58: :: SUPINF_2:58
for X being set
for F being PartFunc of X,ExtREAL holds
( F is nonnegative iff for n being Element of X holds 0. <= F . n )
proof end;

theorem Th59: :: SUPINF_2:59
for F being Function of NAT,ExtREAL
for n being Element of NAT st F is nonnegative holds
( (Ser F) . n <= (Ser F) . (n + 1) & 0. <= (Ser F) . n )
proof end;

theorem Th60: :: SUPINF_2:60
for F being Function of NAT,ExtREAL st F is nonnegative holds
for n, m being Element of NAT holds (Ser F) . n <= (Ser F) . (n + m)
proof end;

theorem Th61: :: SUPINF_2:61
for F1, F2 being Function of NAT,ExtREAL st ( for n being Element of NAT holds F1 . n <= F2 . n ) holds
for n being Element of NAT holds (Ser F1) . n <= (Ser F2) . n
proof end;

theorem Th62: :: SUPINF_2:62
for F1, F2 being Function of NAT,ExtREAL st ( for n being Element of NAT holds F1 . n <= F2 . n ) holds
SUM F1 <= SUM F2
proof end;

theorem Th63: :: SUPINF_2:63
for F being Function of NAT,ExtREAL holds
( (Ser F) . 0 = F . 0 & ( for n being Element of NAT
for y being R_eal st y = (Ser F) . n holds
(Ser F) . (n + 1) = y + (F . (n + 1)) ) )
proof end;

theorem Th64: :: SUPINF_2:64
for F being Function of NAT,ExtREAL st F is nonnegative & ex n being Element of NAT st F . n = +infty holds
SUM F = +infty
proof end;

definition
let F be Function of NAT,ExtREAL;
attr F is summable means :Def24: :: SUPINF_2:def 24
 SUM F in REAL ;
end;

:: deftheorem Def24 defines summable SUPINF_2:def 24 :
for F being Function of NAT,ExtREAL holds
( F is summable iff SUM F in REAL );

theorem :: SUPINF_2:65
for F being Function of NAT,ExtREAL st F is nonnegative & ex n being Element of NAT st F . n = +infty holds
not F is summable
proof end;

theorem :: SUPINF_2:66
for F1, F2 being Function of NAT,ExtREAL st F1 is nonnegative & ( for n being Element of NAT holds F1 . n <= F2 . n ) & F2 is summable holds
F1 is summable
proof end;

theorem Th67: :: SUPINF_2:67
for F being Function of NAT,ExtREAL st F is nonnegative holds
for n being Nat st ( for r being Element of NAT st n <= r holds
F . r = 0. ) holds
SUM F = (Ser F) . n
proof end;

theorem Th68: :: SUPINF_2:68
for F being Function of NAT,ExtREAL st ( for n being Element of NAT holds F . n in REAL ) holds
for n being Element of NAT holds (Ser F) . n in REAL
proof end;

theorem :: SUPINF_2:69
for F being Function of NAT,ExtREAL st F is nonnegative & ex n being Element of NAT st
( ( for k being Element of NAT st n <= k holds
F . k = 0. ) & ( for k being Element of NAT st k <= n holds
F . k <> +infty ) ) holds
F is summable
proof end;

theorem :: SUPINF_2:70
for X being set
for F being PartFunc of X,ExtREAL holds
( F is nonnegative iff for n being set holds 0. <= F . n )
proof end;

theorem :: SUPINF_2:71
for X being set
for F being PartFunc of X,ExtREAL st ( for n being set st n in dom F holds
0. <= F . n ) holds
F is nonnegative
proof end;