:: MESFUNC5 semantic presentation

begin

theorem :: MESFUNC5:1
for x, y being ( ( ) ( ext-real ) R_eal) holds |.(x : ( ( ) ( ext-real ) R_eal) - y : ( ( ) ( ext-real ) R_eal) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) .| : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = |.(y : ( ( ) ( ext-real ) R_eal) - x : ( ( ) ( ext-real ) R_eal) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) .| : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:2
for x, y being ( ( ) ( ext-real ) R_eal) holds y : ( ( ) ( ext-real ) R_eal) - x : ( ( ) ( ext-real ) R_eal) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= |.(x : ( ( ) ( ext-real ) R_eal) - y : ( ( ) ( ext-real ) R_eal) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) .| : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:3
for x, y being ( ( ) ( ext-real ) R_eal)
for e being ( ( real ) ( complex real ext-real ) number ) holds
( not |.(x : ( ( ) ( ext-real ) R_eal) - y : ( ( ) ( ext-real ) R_eal) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) .| : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) < e : ( ( real ) ( complex real ext-real ) number ) or ( x : ( ( ) ( ext-real ) R_eal) = +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) & y : ( ( ) ( ext-real ) R_eal) = +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) or ( x : ( ( ) ( ext-real ) R_eal) = -infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) & y : ( ( ) ( ext-real ) R_eal) = -infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) or ( x : ( ( ) ( ext-real ) R_eal) <> +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) & x : ( ( ) ( ext-real ) R_eal) <> -infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) & y : ( ( ) ( ext-real ) R_eal) <> +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) & y : ( ( ) ( ext-real ) R_eal) <> -infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ) ;

theorem :: MESFUNC5:4
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for p being ( ( ) ( ext-real ) R_eal) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) <= p : ( ( ) ( ext-real ) R_eal) & p : ( ( ) ( ext-real ) R_eal) < n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
ex k being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st
( 1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) <= k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) & k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) <= (2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) |^ n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) ) & (k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) - 1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) ) / (2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) |^ n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) ) <= p : ( ( ) ( ext-real ) R_eal) & p : ( ( ) ( ext-real ) R_eal) < k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) / (2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) |^ n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) ) ) ;

theorem :: MESFUNC5:5
for n, k being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for p being ( ( ) ( ext-real ) R_eal) st k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) <= (2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) |^ n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) ) & n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) <= p : ( ( ) ( ext-real ) R_eal) holds
k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) / (2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) |^ n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) ) <= p : ( ( ) ( ext-real ) R_eal) ;

theorem :: MESFUNC5:6
for x, y, k being ( ( ext-real ) ( ext-real ) number ) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) <= k : ( ( ext-real ) ( ext-real ) number ) holds
( k : ( ( ext-real ) ( ext-real ) number ) * (max (x : ( ( ext-real ) ( ext-real ) number ) ,y : ( ( ext-real ) ( ext-real ) number ) )) : ( ( ) ( ext-real ) set ) : ( ( ext-real ) ( ext-real ) set ) = max ((k : ( ( ext-real ) ( ext-real ) number ) * x : ( ( ext-real ) ( ext-real ) number ) ) : ( ( ext-real ) ( ext-real ) set ) ,(k : ( ( ext-real ) ( ext-real ) number ) * y : ( ( ext-real ) ( ext-real ) number ) ) : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( ext-real ) set ) & k : ( ( ext-real ) ( ext-real ) number ) * (min (x : ( ( ext-real ) ( ext-real ) number ) ,y : ( ( ext-real ) ( ext-real ) number ) )) : ( ( ) ( ext-real ) set ) : ( ( ext-real ) ( ext-real ) set ) = min ((k : ( ( ext-real ) ( ext-real ) number ) * x : ( ( ext-real ) ( ext-real ) number ) ) : ( ( ext-real ) ( ext-real ) set ) ,(k : ( ( ext-real ) ( ext-real ) number ) * y : ( ( ext-real ) ( ext-real ) number ) ) : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( ext-real ) set ) ) ;

theorem :: MESFUNC5:7
for x, y, k being ( ( ) ( ext-real ) R_eal) st k : ( ( ) ( ext-real ) R_eal) <= 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
( k : ( ( ) ( ext-real ) R_eal) * (min (x : ( ( ) ( ext-real ) R_eal) ,y : ( ( ) ( ext-real ) R_eal) )) : ( ( ) ( ext-real ) set ) : ( ( ext-real ) ( ext-real ) set ) = max ((k : ( ( ) ( ext-real ) R_eal) * x : ( ( ) ( ext-real ) R_eal) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ,(k : ( ( ) ( ext-real ) R_eal) * y : ( ( ) ( ext-real ) R_eal) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) : ( ( ) ( ext-real ) set ) & k : ( ( ) ( ext-real ) R_eal) * (max (x : ( ( ) ( ext-real ) R_eal) ,y : ( ( ) ( ext-real ) R_eal) )) : ( ( ) ( ext-real ) set ) : ( ( ext-real ) ( ext-real ) set ) = min ((k : ( ( ) ( ext-real ) R_eal) * x : ( ( ) ( ext-real ) R_eal) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ,(k : ( ( ) ( ext-real ) R_eal) * y : ( ( ) ( ext-real ) R_eal) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) : ( ( ) ( ext-real ) set ) ) ;

begin

definition
let IT be ( ( ) ( ) set ) ;
attr IT is nonpositive means :: MESFUNC5:def 1
 for x being ( ( ) ( ext-real ) R_eal) st x : ( ( ) ( ext-real ) R_eal) in IT : ( ( ) ( ) set ) holds
x : ( ( ) ( ext-real ) R_eal) <= 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) ;
end;

definition
let R be ( ( Relation-like ) ( Relation-like ) Relation) ;
attr R is nonpositive means :: MESFUNC5:def 2
 rng R : ( ( ) ( ) set ) : ( ( ) ( ) set ) is nonpositive ;
end;

theorem :: MESFUNC5:8
for X being ( ( ) ( ) set )
for F being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) holds
( F : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonpositive iff for n being ( ( ) ( ) set ) holds F : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . n : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= 0. : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ;

theorem :: MESFUNC5:9
for X being ( ( ) ( ) set )
for F being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st ( for n being ( ( ) ( ) set ) st n : ( ( ) ( ) set ) in dom F : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) holds
F : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . n : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= 0. : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) holds
F : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonpositive ;

definition
let R be ( ( Relation-like ) ( Relation-like ) Relation) ;
attr R is without-infty means :: MESFUNC5:def 3
 not -infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) in rng R : ( ( ) ( ) set ) : ( ( ) ( ) set ) ;
attr R is without+infty means :: MESFUNC5:def 4
 not +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) in rng R : ( ( ) ( ) set ) : ( ( ) ( ) set ) ;
end;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ;
:: original: without-infty
redefine attr f is without-infty means :: MESFUNC5:def 5
 for x being ( ( ) ( ) set ) holds -infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) < f : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;
:: original: without+infty
redefine attr f is without+infty means :: MESFUNC5:def 6
 for x being ( ( ) ( ) set ) holds f : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) < +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;
end;

theorem :: MESFUNC5:10
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) holds
( ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
-infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) < f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) iff f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is () ) ;

theorem :: MESFUNC5:11
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) holds
( ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) < +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) iff f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is () ) ;

theorem :: MESFUNC5:12
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is () ;

theorem :: MESFUNC5:13
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonpositive holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is () ;

registration
let X be ( ( non empty ) ( non empty ) set ) ;
cluster Function-like nonnegative -> Function-like () for ( ( ) ( ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( V59() ) set ) : ( ( ) ( non empty ) set ) ) ;
cluster Function-like nonpositive -> Function-like () for ( ( ) ( ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( V59() ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

theorem :: MESFUNC5:14
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is () & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is () ) ;

theorem :: MESFUNC5:15
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | Y : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ;

theorem :: MESFUNC5:16
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is () & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is () holds
dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC5:17
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is () & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is () holds
dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC5:18
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for F being ( ( Function-like V32( RAT : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered rational-membered V74() ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ( Relation-like RAT : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered rational-membered V74() ) set ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V32( RAT : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered rational-membered V74() ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) Function of RAT : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered rational-membered V74() ) set ) ,S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for r being ( ( ) ( complex real ext-real ) Real)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is () & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is () & ( for p being ( ( rational ) ( complex real ext-real rational ) Rational) holds F : ( ( Function-like V32( RAT : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered rational-membered V74() ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ( Relation-like RAT : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered rational-membered V74() ) set ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V32( RAT : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered rational-membered V74() ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) Function of RAT : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered rational-membered V74() ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . p : ( ( rational ) ( complex real ext-real rational ) Rational) : ( ( ) ( ) set ) = (A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ (less_dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ,(R_EAL p : ( ( rational ) ( complex real ext-real rational ) Rational) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) )) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ (less_dom (g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ,(R_EAL (r : ( ( ) ( complex real ext-real ) Real) - p : ( ( rational ) ( complex real ext-real rational ) Rational) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) )) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) holds
A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ (less_dom ((f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ,(R_EAL r : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) )) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = union (rng F : ( ( Function-like V32( RAT : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered rational-membered V74() ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ( Relation-like RAT : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered rational-membered V74() ) set ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V32( RAT : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered rational-membered V74() ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) Function of RAT : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered rational-membered V74() ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) -valued Function-like V58() V59() V60() ) PartFunc of ,) ;
func R_EAL f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) equals :: MESFUNC5:def 7
f : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

theorem :: MESFUNC5:19
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ;

theorem :: MESFUNC5:20
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for c being ( ( ) ( complex real ext-real ) Real) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative holds
( ( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) <= c : ( ( ) ( complex real ext-real ) Real) implies c : ( ( ) ( complex real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ) & ( c : ( ( ) ( complex real ext-real ) Real) <= 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) implies c : ( ( ) ( complex real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is nonpositive ) ) ;

theorem :: MESFUNC5:21
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
( g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) & -infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) < g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) < +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ;

theorem :: MESFUNC5:22
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative holds
( dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ) ;

theorem :: MESFUNC5:23
for X being ( ( non empty ) ( non empty ) set )
for f, g, h being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative & h : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative holds
( dom ((f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) + h : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = ((dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom h : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) + h : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in ((dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom h : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
((f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) + h : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = ((f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) + (g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) + (h : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ) ;

theorem :: MESFUNC5:24
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is () & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is () holds
( dom ((max+ (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) + (max- f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & dom ((max- (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) + (max+ f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & dom (((max+ (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) + (max- f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) + (max- g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & dom (((max- (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) + (max+ f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) + (max+ g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & (max+ (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) + (max- f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative & (max- (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) + (max+ f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ) ;

theorem :: MESFUNC5:25
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is () & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is () & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is () & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is () holds
((max+ (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) + (max- f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) + (max- g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) = ((max- (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) + (max+ f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) + (max+ g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC5:26
for C being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for c being ( ( ) ( complex real ext-real ) Real) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) <= c : ( ( ) ( complex real ext-real ) Real) holds
( max+ (c : ( ( ) ( complex real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) = c : ( ( ) ( complex real ext-real ) Real) (#) (max+ f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) & max- (c : ( ( ) ( complex real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) = c : ( ( ) ( complex real ext-real ) Real) (#) (max- f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC5:27
for C being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for c being ( ( ) ( complex real ext-real ) Real) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) <= c : ( ( ) ( complex real ext-real ) Real) holds
( max+ ((- c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) ) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) = c : ( ( ) ( complex real ext-real ) Real) (#) (max- f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) & max- ((- c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) ) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) = c : ( ( ) ( complex real ext-real ) Real) (#) (max+ f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC5:28
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for A being ( ( ) ( ) set ) holds
( max+ (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | A : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) = (max+ f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) | A : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) & max- (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | A : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) = (max- f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) | A : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC5:29
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for B being ( ( ) ( ) set ) st B : ( ( ) ( ) set ) c= dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
( dom ((f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) | B : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = B : ( ( ) ( ) set ) & dom ((f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | B : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) + (g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | B : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = B : ( ( ) ( ) set ) & (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) | B : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) = (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | B : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) + (g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | B : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC5:30
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for a being ( ( ) ( ext-real ) R_eal) holds eq_dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ,a : ( ( ) ( ext-real ) R_eal) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) " {a : ( ( ) ( ext-real ) R_eal) } : ( ( ) ( non empty finite ext-real-membered left_end right_end ) set ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

begin

theorem :: MESFUNC5:31
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is () & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is () & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC5:32
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = {} : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) set ) holds
ex F being ( ( V113() ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V113() ) Finite_Sep_Sequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ex a, x being ( ( ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() ) FinSequence of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) st
( F : ( ( V113() ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V113() ) Finite_Sep_Sequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,a : ( ( ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() ) FinSequence of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) are_Re-presentation_of f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) & a : ( ( ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() ) FinSequence of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) . 1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st 2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural non empty complex real ext-real positive non negative integer rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) <= n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) & n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) in dom a : ( ( ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() ) FinSequence of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) holds
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) < a : ( ( ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() ) FinSequence of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) & a : ( ( ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() ) FinSequence of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) < +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ) & dom x : ( ( ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() ) FinSequence of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) = dom F : ( ( V113() ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V113() ) Finite_Sep_Sequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) in dom x : ( ( ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() ) FinSequence of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) holds
x : ( ( ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() ) FinSequence of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = (a : ( ( ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() ) FinSequence of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) * ((M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * F : ( ( V113() ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V113() ) Finite_Sep_Sequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() ) FinSequence of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) & Sum x : ( ( ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() ) FinSequence of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: MESFUNC5:33
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for r, s being ( ( ) ( complex real ext-real ) Real) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
(A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ (great_eq_dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ,(R_EAL r : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) )) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (less_dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ,(R_EAL s : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) )) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: MESFUNC5:34
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: MESFUNC5:35
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for F being ( ( V113() ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V113() ) Finite_Sep_Sequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for G being ( ( Relation-like Function-like FinSequence-like ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like ) FinSequence) st dom F : ( ( V113() ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V113() ) Finite_Sep_Sequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) = dom G : ( ( Relation-like Function-like FinSequence-like ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like ) FinSequence) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) in dom F : ( ( V113() ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V113() ) Finite_Sep_Sequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) holds
G : ( ( Relation-like Function-like FinSequence-like ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like ) FinSequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ) set ) = (F : ( ( V113() ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V113() ) Finite_Sep_Sequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( ) set ) /\ A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) holds
G : ( ( Relation-like Function-like FinSequence-like ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like ) FinSequence) is ( ( V113() ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V113() ) Finite_Sep_Sequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC5:36
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for F, G being ( ( V113() ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V113() ) Finite_Sep_Sequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for a being ( ( ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() ) FinSequence of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) st dom F : ( ( V113() ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V113() ) Finite_Sep_Sequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) = dom G : ( ( V113() ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V113() ) Finite_Sep_Sequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) in dom F : ( ( V113() ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V113() ) Finite_Sep_Sequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) holds
G : ( ( V113() ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V113() ) Finite_Sep_Sequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ) set ) = (F : ( ( V113() ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V113() ) Finite_Sep_Sequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( ) set ) /\ A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) & F : ( ( V113() ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V113() ) Finite_Sep_Sequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,a : ( ( ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() ) FinSequence of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) are_Re-presentation_of f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) holds
G : ( ( V113() ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V113() ) Finite_Sep_Sequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,a : ( ( ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() ) FinSequence of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) are_Re-presentation_of f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC5:37
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) holds
dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC5:38
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: MESFUNC5:39
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for c being ( ( ) ( complex real ext-real ) Real) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) holds
c : ( ( ) ( complex real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: MESFUNC5:40
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ;

theorem :: MESFUNC5:41
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for c being ( ( ) ( ext-real ) R_eal) st c : ( ( ) ( ext-real ) R_eal) <> +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) & c : ( ( ) ( ext-real ) R_eal) <> -infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) holds
ex f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st
( f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = c : ( ( ) ( ext-real ) R_eal) ) ) ;

theorem :: MESFUNC5:42
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for B, BF being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & BF : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on BF : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC5:43
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is () & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is () holds
(max+ (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) + (max- f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC5:44
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is () & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is () holds
(max- (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) + (max+ f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC5:45
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for A being ( ( ) ( ) set ) st A : ( ( ) ( ) set ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) holds
0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) <= M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . A : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:46
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st ex E1 being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( E1 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on E1 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & ex E2 being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( E2 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on E2 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) " {+infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) } : ( ( ) ( non empty finite ext-real-membered left_end right_end ) set ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) " {-infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) } : ( ( ) ( non empty finite ext-real-membered left_end right_end ) set ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) " {+infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) } : ( ( ) ( non empty finite ext-real-membered left_end right_end ) set ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) " {-infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) } : ( ( ) ( non empty finite ext-real-membered left_end right_end ) set ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) holds
dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: MESFUNC5:47
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st ex E1 being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( E1 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on E1 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & ex E2 being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( E2 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on E2 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) holds
ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ;

theorem :: MESFUNC5:48
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for A, B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) iff f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) M13(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )) ) ;

theorem :: MESFUNC5:49
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st ex A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = A : ( ( ) ( complex real ext-real ) Real) holds
for c being ( ( ) ( complex real ext-real ) Real)
for B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
c : ( ( ) ( complex real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

begin

definition
mode ExtREAL_sequence is ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) Function of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;
end;

definition
let seq be ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ;
attr seq is convergent_to_finite_number means :: MESFUNC5:def 8
 ex g being ( ( real ) ( complex real ext-real ) number ) st
for p being ( ( real ) ( complex real ext-real ) number ) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) < p : ( ( real ) ( complex real ext-real ) number ) holds
ex n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st
for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) <= m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
|.((seq : ( ( ) ( ) set ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) - (R_EAL g : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) .| : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) < p : ( ( real ) ( complex real ext-real ) number ) ;
end;

definition
let seq be ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ;
attr seq is convergent_to_+infty means :: MESFUNC5:def 9
 for g being ( ( real ) ( complex real ext-real ) number ) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) < g : ( ( real ) ( complex real ext-real ) number ) holds
ex n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st
for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) <= m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
g : ( ( real ) ( complex real ext-real ) number ) <= seq : ( ( ) ( ) set ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;
end;

definition
let seq be ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ;
attr seq is convergent_to_-infty means :: MESFUNC5:def 10
 for g being ( ( real ) ( complex real ext-real ) number ) st g : ( ( real ) ( complex real ext-real ) number ) < 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
ex n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st
for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) <= m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
seq : ( ( ) ( ) set ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= g : ( ( real ) ( complex real ext-real ) number ) ;
end;

theorem :: MESFUNC5:50
for seq being ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) st seq : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is convergent_to_+infty holds
( not seq : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is convergent_to_-infty & not seq : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is convergent_to_finite_number ) ;

theorem :: MESFUNC5:51
for seq being ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) st seq : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is convergent_to_-infty holds
( not seq : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is convergent_to_+infty & not seq : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is convergent_to_finite_number ) ;

definition
let seq be ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ;
attr seq is convergent means :: MESFUNC5:def 11
( seq : ( ( ) ( ) set ) is convergent_to_finite_number or seq : ( ( ) ( ) set ) is convergent_to_+infty or seq : ( ( ) ( ) set ) is convergent_to_-infty );
end;

definition
let seq be ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ;
assume seq : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is convergent ;
func lim seq -> ( ( ) ( ext-real ) R_eal) means :: MESFUNC5:def 12
( ex g being ( ( real ) ( complex real ext-real ) number ) st
( it : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool seq : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = g : ( ( real ) ( complex real ext-real ) number ) & ( for p being ( ( real ) ( complex real ext-real ) number ) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) < p : ( ( real ) ( complex real ext-real ) number ) holds
ex n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st
for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) <= m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
|.((seq : ( ( ) ( ) set ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) - it : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool seq : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) .| : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) < p : ( ( real ) ( complex real ext-real ) number ) ) & seq : ( ( ) ( ) set ) is convergent_to_finite_number ) or ( it : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool seq : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) & seq : ( ( ) ( ) set ) is convergent_to_+infty ) or ( it : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool seq : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = -infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) & seq : ( ( ) ( ) set ) is convergent_to_-infty ) );
end;

theorem :: MESFUNC5:52
for seq being ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence)
for r being ( ( real ) ( complex real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds seq : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = r : ( ( real ) ( complex real ext-real ) number ) ) holds
( seq : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is convergent_to_finite_number & lim seq : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) : ( ( ) ( ext-real ) R_eal) = r : ( ( real ) ( complex real ext-real ) number ) ) ;

theorem :: MESFUNC5:53
for F being ( ( ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() ) FinSequence of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) in dom F : ( ( ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() ) FinSequence of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) holds
0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) <= F : ( ( ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() ) FinSequence of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) holds
0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) <= Sum F : ( ( ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V59() ) FinSequence of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:54
for L being ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) st ( for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) <= m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) holds
( L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is convergent & lim L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) : ( ( ) ( ext-real ) R_eal) = sup (rng L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ) : ( ( non empty V93() ) ( non empty ext-real-membered V93() ) Element of bool ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ;

theorem :: MESFUNC5:55
for L, G being ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= G : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) holds
sup (rng L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ) : ( ( non empty V93() ) ( non empty ext-real-membered V93() ) Element of bool ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= sup (rng G : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ) : ( ( non empty V93() ) ( non empty ext-real-membered V93() ) Element of bool ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:56
for L being ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence)
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= sup (rng L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ) : ( ( non empty V93() ) ( non empty ext-real-membered V93() ) Element of bool ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:57
for L being ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence)
for K being ( ( ) ( ext-real ) R_eal) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= K : ( ( ) ( ext-real ) R_eal) ) holds
sup (rng L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ) : ( ( non empty V93() ) ( non empty ext-real-membered V93() ) Element of bool ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= K : ( ( ) ( ext-real ) R_eal) ;

theorem :: MESFUNC5:58
for L being ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence)
for K being ( ( ) ( ext-real ) R_eal) st K : ( ( ) ( ext-real ) R_eal) <> +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= K : ( ( ) ( ext-real ) R_eal) ) holds
sup (rng L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ) : ( ( non empty V93() ) ( non empty ext-real-membered V93() ) Element of bool ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) < +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:59
for L being ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) st L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is () holds
( sup (rng L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ) : ( ( non empty V93() ) ( non empty ext-real-membered V93() ) Element of bool ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <> +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) iff ex K being ( ( real ) ( complex real ext-real ) number ) st
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) < K : ( ( real ) ( complex real ext-real ) number ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= K : ( ( real ) ( complex real ext-real ) number ) ) ) ) ;

theorem :: MESFUNC5:60
for L being ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence)
for c being ( ( ext-real ) ( ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = c : ( ( ext-real ) ( ext-real ) number ) ) holds
( L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is convergent & lim L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) : ( ( ) ( ext-real ) R_eal) = c : ( ( ext-real ) ( ext-real ) number ) & lim L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) : ( ( ) ( ext-real ) R_eal) = sup (rng L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ) : ( ( non empty V93() ) ( non empty ext-real-membered V93() ) Element of bool ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ;

theorem :: MESFUNC5:61
for J, K, L being ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) st ( for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) <= m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
J : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= J : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) & ( for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) <= m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
K : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= K : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) & J : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is () & K : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is () & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (J : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) + (K : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) holds
( L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is convergent & lim L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) : ( ( ) ( ext-real ) R_eal) = sup (rng L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ) : ( ( non empty V93() ) ( non empty ext-real-membered V93() ) Element of bool ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) & lim L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) : ( ( ) ( ext-real ) R_eal) = (lim J : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ) : ( ( ) ( ext-real ) R_eal) + (lim K : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ) : ( ( ) ( ext-real ) R_eal) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) & sup (rng L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ) : ( ( non empty V93() ) ( non empty ext-real-membered V93() ) Element of bool ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = (sup (rng K : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ) : ( ( non empty V93() ) ( non empty ext-real-membered V93() ) Element of bool ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) + (sup (rng J : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ) : ( ( non empty V93() ) ( non empty ext-real-membered V93() ) Element of bool ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ;

theorem :: MESFUNC5:62
for L, K being ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence)
for c being ( ( ) ( complex real ext-real ) Real) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) <= c : ( ( ) ( complex real ext-real ) Real) & L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is () & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds K : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = (R_EAL c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) * (L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) holds
( sup (rng K : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ) : ( ( non empty V93() ) ( non empty ext-real-membered V93() ) Element of bool ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = (R_EAL c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) * (sup (rng L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ) : ( ( non empty V93() ) ( non empty ext-real-membered V93() ) Element of bool ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) & K : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is () ) ;

theorem :: MESFUNC5:63
for L, K being ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence)
for c being ( ( ) ( complex real ext-real ) Real) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) <= c : ( ( ) ( complex real ext-real ) Real) & ( for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) <= m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds K : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = (R_EAL c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) * (L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) & L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is () holds
( ( for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) <= m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
K : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= K : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) & K : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is () & K : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is convergent & lim K : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) : ( ( ) ( ext-real ) R_eal) = sup (rng K : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ) : ( ( non empty V93() ) ( non empty ext-real-membered V93() ) Element of bool ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) & lim K : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) : ( ( ) ( ext-real ) R_eal) = (R_EAL c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) * (lim L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ) : ( ( ) ( ext-real ) R_eal) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ;

begin

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let H be ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) ;
let x be ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) ;
func H # x -> ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) means :: MESFUNC5:def 13
 for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds it : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = (H : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( V59() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( H : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ,K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined H : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ,K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ,K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;
end;

definition
let D1, D2 be ( ( ) ( ) set ) ;
let F be ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (D1 : ( ( ) ( ) set ) ,D2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of D1 : ( ( ) ( ) set ) ,D2 : ( ( ) ( ) set ) ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (D1 : ( ( ) ( ) set ) ,D2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of D1 : ( ( ) ( ) set ) ,D2 : ( ( ) ( ) set ) ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (D1 : ( ( ) ( ) set ) ,D2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of D1 : ( ( ) ( ) set ) ,D2 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (D1 : ( ( ) ( ) set ) ,D2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) PartFunc-set of D1 : ( ( ) ( ) set ) ,D2 : ( ( ) ( ) set ) ) ) ;
let n be ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ;
:: original: .
redefine func F . n -> ( ( Function-like ) ( Relation-like D1 : ( ( ) ( ) set ) -defined D2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool D1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like ) PartFunc of ,) ;
end;

theorem :: MESFUNC5:64
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st ex A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( A : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on A : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative holds
ex F being ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) st
( ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
( F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & dom (F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative ) & ( for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <= m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
(F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= (F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) & ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
( F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is convergent & lim (F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) : ( ( ) ( ext-real ) R_eal) = f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ) ) ;

begin

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ;
func integral' (M,f) -> ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) equals :: MESFUNC5:def 14
 integral (X : ( ( ) ( ) set ) ,S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ,K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ,K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ,K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,f : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) if dom f : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) <> {} : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) set )
otherwise 0. : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;
end;

theorem :: MESFUNC5:65
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative holds
( dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = (integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | (dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) + (integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | (dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ;

theorem :: MESFUNC5:66
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for c being ( ( ) ( complex real ext-real ) Real) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative & 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) <= c : ( ( ) ( complex real ext-real ) Real) holds
integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(c : ( ( ) ( complex real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = (R_EAL c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) * (integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:67
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for A, B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative & A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) misses B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | (A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) \/ B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) M13(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = (integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) + (integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:68
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative holds
0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) <= integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:69
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) holds
( dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | (dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = (integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) + (integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | (dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ;

theorem :: MESFUNC5:70
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) holds
integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | (dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | (dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:71
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for c being ( ( ) ( ext-real ) R_eal) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) <= c : ( ( ) ( ext-real ) R_eal) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = c : ( ( ) ( ext-real ) R_eal) ) holds
integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = c : ( ( ) ( ext-real ) R_eal) * (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:72
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative holds
integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | (eq_dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ,(R_EAL 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) )) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC5:73
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative holds
integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC5:74
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for F being ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for L being ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) st g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & ( for x being ( ( ) ( ) set ) st x : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) in dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) < g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds dom (F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative ) & ( for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) <= m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
(F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= (F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) & ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) in dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
( F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is convergent & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= lim (F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) : ( ( ) ( ext-real ) R_eal) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) holds
( L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is convergent & integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= lim L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) : ( ( ) ( ext-real ) R_eal) ) ;

theorem :: MESFUNC5:75
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for F being ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) st g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds dom (F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative ) & ( for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) <= m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
(F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= (F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) & ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) in dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
( F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is convergent & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= lim (F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ) : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) : ( ( ) ( ext-real ) R_eal) ) ) holds
ex G being ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) st
( ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds G : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) & G : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is convergent & sup (rng G : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) ) : ( ( non empty V93() ) ( non empty ext-real-membered V93() ) Element of bool ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = lim G : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) : ( ( ) ( ext-real ) R_eal) & integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= lim G : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) : ( ( ) ( ext-real ) R_eal) ) ;

theorem :: MESFUNC5:76
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for F, G being ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for K, L being ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
( F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & dom (F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative ) & ( for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) <= m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
(F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= (F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
( G : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & dom (G : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds G : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative ) & ( for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) <= m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
(G : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= (G : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) & ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) in A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is convergent & G : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is convergent & lim (F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) : ( ( ) ( ext-real ) R_eal) = lim (G : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) : ( ( ) ( ext-real ) R_eal) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
( K : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) & L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(G : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ) holds
( K : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is convergent & L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is convergent & lim K : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) : ( ( ) ( ext-real ) R_eal) = lim L : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) : ( ( ) ( ext-real ) R_eal) ) ;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ;
assume that
ex A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) st
( A : ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool X : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ) and
f : ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative ;
func integral+ (M,f) -> ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) means :: MESFUNC5:def 15
 ex F being ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( ) ( ) set ) ) ex K being ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) st
( ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
( F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & dom (F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) = dom f : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative ) & ( for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) <= m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
for x being ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) st x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) in dom f : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) holds
(F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= (F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) & ( for x being ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) st x : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) in dom f : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) holds
( F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) # x : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is convergent & lim (F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) # x : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) : ( ( ) ( ext-real ) R_eal) = f : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) . x : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds K : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = integral' (M : ( ( S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ,K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ,K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ,K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(F : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) & K : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) is convergent & it : ( ( ) ( ) set ) = lim K : ( ( Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() ) ExtREAL_sequence) : ( ( ) ( ext-real ) R_eal) );
end;

theorem :: MESFUNC5:77
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative holds
integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:78
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st ex A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & ex B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative holds
ex C being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( C : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = (integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | C : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) + (integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | C : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ;

theorem :: MESFUNC5:79
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st ex A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative holds
0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) <= integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:80
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative holds
0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) <= integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:81
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for A, B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative & A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) misses B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | (A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) \/ B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) M13(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = (integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) + (integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:82
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative & M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC5:83
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for A, B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative & A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:84
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for E, A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative & E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | (E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) \ A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) M13(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:85
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( E : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & E : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) = dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative & ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) holds
integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:86
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for c being ( ( ) ( complex real ext-real ) Real) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) <= c : ( ( ) ( complex real ext-real ) Real) & ex A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative holds
integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(c : ( ( ) ( complex real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = (R_EAL c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) * (integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:87
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st ex A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( A : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) & ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) = f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) holds
integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

begin

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ;
func Integral (M,f) -> ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) equals :: MESFUNC5:def 16
(integral+ (M : ( ( S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ,K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ,K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ,K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(max+ f : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( V59() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) - (integral+ (M : ( ( S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ,K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ,K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ,K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(max- f : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( V59() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;
end;

theorem :: MESFUNC5:88
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st ex A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative holds
Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:89
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative holds
( Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) & Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ;

theorem :: MESFUNC5:90
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st ex A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative holds
0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) <= Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:91
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for A, B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative & A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) misses B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | (A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) \/ B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) M13(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = (Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) + (Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:92
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative holds
0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) <= Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:93
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for A, B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative & A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:94
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC5:95
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for E, A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | (E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) \ A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) M13(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ;
pred f is_integrable_on M means :: MESFUNC5:def 17
( ex A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) st
( A : ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) is_measurable_on A : ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ) & integral+ (M : ( ( S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ,K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ,K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ,K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(max+ f : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) < +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) & integral+ (M : ( ( S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ,K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ,K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ,K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(max- f : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) < +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) );
end;

theorem :: MESFUNC5:96
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) <= integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(max+ f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) & 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) <= integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(max- f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) & -infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) < Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) & Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) < +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ;

theorem :: MESFUNC5:97
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(max+ (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(max+ f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) & integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(max- (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(max- f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ;

theorem :: MESFUNC5:98
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for A, B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) misses B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | (A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) \/ B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) M13(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = (Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) + (Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:99
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for A, B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) \ A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = (Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) + (Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ;

theorem :: MESFUNC5:100
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st ex A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) holds
( f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) iff |.f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) .| : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ;

theorem :: MESFUNC5:101
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
|.(Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) .| : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,|.f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) .| : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:102
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st ex A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( A : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) & dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
|.(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) .| : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,|.f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) .| : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) <= Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ;

theorem :: MESFUNC5:103
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for r being ( ( ) ( complex real ext-real ) Real) st dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) <= r : ( ( ) ( complex real ext-real ) Real) & dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) <> {} : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) set ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = r : ( ( ) ( complex real ext-real ) Real) ) holds
integral (X : ( ( non empty ) ( non empty ) set ) ,S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = r : ( ( ) ( complex real ext-real ) Real) * (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ext-real ) ( ext-real ) set ) ;

theorem :: MESFUNC5:104
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for r being ( ( ) ( complex real ext-real ) Real) st dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) <= r : ( ( ) ( complex real ext-real ) Real) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = r : ( ( ) ( complex real ext-real ) Real) ) holds
integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = (R_EAL r : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) * (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;

theorem :: MESFUNC5:105
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) " {+infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) } : ( ( ) ( non empty finite ext-real-membered left_end right_end ) set ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) " {-infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) } : ( ( ) ( non empty finite ext-real-membered left_end right_end ) set ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) " {+infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) } : ( ( ) ( non empty finite ext-real-membered left_end right_end ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) & M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) " {-infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) } : ( ( ) ( non empty finite ext-real-membered left_end right_end ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) & (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) " {+infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) } : ( ( ) ( non empty finite ext-real-membered left_end right_end ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) \/ (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) " {-infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) } : ( ( ) ( non empty finite ext-real-membered left_end right_end ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . ((f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) " {+infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) } : ( ( ) ( non empty finite ext-real-membered left_end right_end ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) \/ (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) " {-infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) } : ( ( ) ( non empty finite ext-real-membered left_end right_end ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural empty complex real ext-real non positive non negative integer functional finite V41() FinSequence-membered rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: MESFUNC5:106
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is nonnegative holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC5:107
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: MESFUNC5:108
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC5:109
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = (Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) + (Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ;

theorem :: MESFUNC5:110
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for c being ( ( ) ( complex real ext-real ) Real) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( c : ( ( ) ( complex real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(c : ( ( ) ( complex real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = (R_EAL c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) * (Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ;
let B be ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
func Integral_on (M,B,f) -> ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) equals :: MESFUNC5:def 18
 Integral (M : ( ( S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ,K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( Relation-like NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) -defined S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like V32( NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ,K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( epsilon-transitive epsilon-connected ordinal non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V74() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite complex-membered ext-real-membered real-membered V74() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty ) set ) ) ,K403(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,(f : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) | B : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ;
end;

theorem :: MESFUNC5:111
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & Integral_on (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = (Integral_on (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) + (Integral_on (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ;

theorem :: MESFUNC5:112
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,)
for c being ( ( ) ( complex real ext-real ) Real)
for B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) | B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & Integral_on (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(c : ( ( ) ( complex real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) :] : ( ( ) ( non empty V59() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) = (R_EAL c : ( ( ) ( complex real ext-real ) Real) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) * (Integral_on (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V67() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) V59() V67() nonnegative sigma-additive () ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V109() V110() V111() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) -valued Function-like V59() ) PartFunc of ,) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered V120() ) set ) ) ) ;