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

begin

notation

synonym 0. for 0 ;

end;

definition

canceled;
:: original: 0.
redefine func 0. -> R_eal;
coherence
0. is R_eal by XXREAL_0:def 1;

end;

:: deftheorem SUPINF_2:def 1 :
canceled;

definition

let x, y be R_eal;
:: 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;

definition

let x be R_eal;
:: original: -
redefine func - x -> R_eal;
coherence
- x is R_eal by XXREAL_0:def 1;

end;

definition

let x, y be R_eal;
:: original: -
redefine func x - y -> R_eal;
coherence
x - y is R_eal by XXREAL_0:def 1;

end;

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;

definition

canceled;
canceled;
canceled;
let X, Y be non empty Subset of ExtREAL;
func X + Y -> Subset of ExtREAL means :Def5: :: SUPINF_2:def 5
for z being R_eal holds
( z in it iff ex x, y being R_eal st
( x in X & y in Y & z = x + y ) );
existence
ex b₁ being Subset of ExtREAL st
for z being R_eal holds
( z in b₁ iff ex x, y being R_eal st
( x in X & y in Y & z = x + y ) )

proof end;

uniqueness
for b₁, b₂ being Subset of ExtREAL st ( for z being R_eal holds
( z in b₁ iff ex x, y being R_eal st
( x in X & y in Y & z = x + y ) ) ) & ( for z being R_eal holds
( z in b₂ iff ex x, y being R_eal st
( x in X & y in Y & z = x + y ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem SUPINF_2:def 2 :
canceled;

:: deftheorem SUPINF_2:def 3 :
canceled;

:: deftheorem SUPINF_2:def 4 :
canceled;

:: deftheorem Def5 defines + SUPINF_2:def 5 :
for X, Y being non empty Subset of ExtREAL
for b₃ being Subset of ExtREAL holds
( b₃ = X + Y iff for z being R_eal holds
( z in b₃ iff ex x, y being R_eal st
( x in X & y in Y & z = x + y ) ) );

registration

let X, Y be non empty Subset of ExtREAL;
cluster X + Y -> non empty ;
coherence
not X + Y is empty

proof end;

end;

definition

let X be non empty Subset of ExtREAL;
func - X -> Subset of ExtREAL means :Def6: :: SUPINF_2:def 6
for z being R_eal holds
( z in it iff ex x being R_eal st
( x in X & z = - x ) );
existence
ex b₁ being Subset of ExtREAL st
for z being R_eal holds
( z in b₁ iff ex x being R_eal st
( x in X & z = - x ) )

proof end;

uniqueness
for b₁, b₂ being Subset of ExtREAL st ( for z being R_eal holds
( z in b₁ iff ex x being R_eal st
( x in X & z = - x ) ) ) & ( for z being R_eal holds
( z in b₂ iff ex x being R_eal st
( x in X & z = - x ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def6 defines - SUPINF_2:def 6 :
for X being non empty Subset of ExtREAL
for b₂ being Subset of ExtREAL holds
( b₂ = - X iff for z being R_eal holds
( z in b₂ iff ex x being R_eal st
( x in X & z = - x ) ) );

registration

let X be non empty Subset of ExtREAL;
cluster - X -> non empty ;
coherence
not - X is empty

proof end;

end;

theorem :: SUPINF_2:21

canceled;

theorem Th22: :: SUPINF_2:22

for X being non empty Subset of ExtREAL holds - (- X) = X

proof end;

theorem Th23: :: SUPINF_2:23

for X being non empty Subset of ExtREAL
for z being R_eal holds
( z in X iff - z in - X )

proof end;

theorem :: SUPINF_2:24

for X, Y being non empty Subset of ExtREAL holds
( X c= Y iff - X c= - Y )

proof end;

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 X be non empty set ;
let Y be non empty Subset of ExtREAL;
let F be Function of X,Y;
func sup F -> R_eal equals :: SUPINF_2:def 7
sup (rng F);
coherence
sup (rng F) is R_eal ;
func inf F -> R_eal equals :: SUPINF_2:def 8
inf (rng F);
coherence
inf (rng F) is R_eal ;

end;

:: deftheorem defines sup SUPINF_2:def 7 :
for X being non empty set
for Y being non empty Subset of ExtREAL
for F being Function of X,Y holds sup F = sup (rng F);

:: deftheorem defines inf SUPINF_2:def 8 :
for X being non empty set
for Y being non empty Subset of ExtREAL
for F being Function 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 non empty set ;
let Y be non empty 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 non empty set
for Y being non empty 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 non empty set
for Y being non empty Subset of ExtREAL
for F being Function of X,Y holds
( F is bounded_below iff -infty < inf F );

definition

let X be non empty set ;
let Y be non empty 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 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_above & F is bounded_below ) );

registration

let X be non empty set ;
let Y be non empty 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

ex F being Function of NAT,ExtREAL st
( F is one-to-one & NAT = rng F & rng F 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 V74() 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 V74() 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)) );

Lm1: for F being Function of NAT,ExtREAL holds rng F is non empty Subset of ExtREAL
;

definition

let F be Function of NAT,ExtREAL;
:: original: proj2
redefine func rng F -> non empty Subset of ExtREAL;
coherence
proj2 F is non empty Subset of ExtREAL by Lm1;

end;

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;