:: MESFUN6C semantic presentation

begin

theorem :: MESFUN6C:1
for a, b being ( ( real ) ( complex real ext-real ) number ) holds
( (R_EAL a : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) + (R_EAL b : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) = a : ( ( real ) ( complex real ext-real ) number ) + b : ( ( real ) ( complex real ext-real ) number ) : ( ( ) ( complex real ext-real ) set ) & - (R_EAL a : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) = - a : ( ( real ) ( complex real ext-real ) number ) : ( ( complex ) ( complex real ext-real ) set ) & (R_EAL a : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) - (R_EAL b : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) = a : ( ( real ) ( complex real ext-real ) number ) - b : ( ( real ) ( complex real ext-real ) number ) : ( ( ) ( complex real ext-real ) set ) & (R_EAL a : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) * (R_EAL b : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) = a : ( ( real ) ( complex real ext-real ) number ) * b : ( ( real ) ( complex real ext-real ) number ) : ( ( ) ( complex real ext-real ) 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 V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ;
let E be ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
pred f is_measurable_on E means :: MESFUN6C:def 1
( Re f : ( ( Function-like V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) Element of K6(K7(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V51() ) set ) ) : ( ( ) ( non empty Relation-like V34() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) & Im f : ( ( Function-like V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) Element of K6(K7(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V51() ) set ) ) : ( ( ) ( non empty Relation-like V34() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) );
end;

theorem :: MESFUN6C:2
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,)
for r being ( ( ) ( complex real ext-real ) Real) holds
( r : ( ( ) ( complex real ext-real ) Real) (#) (Re f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) = Re (r : ( ( ) ( complex real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) & r : ( ( ) ( complex real ext-real ) Real) (#) (Im f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) = Im (r : ( ( ) ( complex real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUN6C:3
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,)
for c being ( ( complex ) ( complex ) number ) holds
( Re (c : ( ( complex ) ( complex ) number ) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) = ((Re c : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) (#) (Re f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) - ((Im c : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) (#) (Im f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) & Im (c : ( ( complex ) ( complex ) number ) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) = ((Im c : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) (#) (Re f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) + ((Re c : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) (#) (Im f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUN6C:4
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) holds
( - (Im f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) = Re (<i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) & Re f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) = Im (<i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUN6C:5
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) holds
( Re (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) = (Re f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) + (Re g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) & Im (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) = (Im f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) + (Im g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUN6C:6
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) holds
( Re (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) = (Re f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) - (Re g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) & Im (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) = (Im f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) - (Im g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUN6C:7
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( (Re f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b4 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) = Re (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b4 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) & (Im f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b4 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) = Im (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b4 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUN6C:8
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) holds f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) = (Re f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) + (<i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) (#) (Im f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUN6C:9
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,)
for B, A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_measurable_on B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUN6C:10
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,)
for A, B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_measurable_on B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) M11(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )) ;

theorem :: MESFUN6C:11
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUN6C:12
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUN6C:13
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( ) ( ) set )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) st Y : ( ( ) ( ) set ) c= dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) holds
( dom ((f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) | Y : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( ) ( ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) + (g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) | Y : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( ) ( ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) = Y : ( ( ) ( ) set ) & (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) | Y : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( ) ( ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) = (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) | Y : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( ) ( ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) + (g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) | Y : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( ) ( ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUN6C:14
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,)
for B, A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_measurable_on B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( ) ( ) Element of K6(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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) | B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b4 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUN6C:15
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) st dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) holds
dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: MESFUN6C:16
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,)
for A, B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) : ( ( ) ( ) Element of K6(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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_measurable_on B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) iff f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) M11(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )) ) ;

theorem :: MESFUN6C:17
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,)
for c being ( ( complex ) ( complex ) number )
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) holds
c : ( ( complex ) ( complex ) number ) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUN6C:18
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) st ex A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) = A : ( ( complex ) ( complex ) number ) holds
for c being ( ( complex ) ( complex ) number )
for B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_measurable_on B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
c : ( ( complex ) ( complex ) number ) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) is_measurable_on B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( 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 V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ;
pred f is_integrable_on M means :: MESFUN6C:def 2
( Re f : ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) Element of K6(K7(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V51() ) set ) ) : ( ( ) ( non empty Relation-like V34() ) set ) ) : ( ( ) ( non empty ) set ) ) & Im f : ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) Element of K6(K7(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V51() ) set ) ) : ( ( ) ( non empty Relation-like V34() ) set ) ) : ( ( ) ( non empty ) set ) ) );
end;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ;
assume f : ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
func Integral (M,f) -> ( ( complex ) ( complex ) number ) means :: MESFUN6C:def 3
 ex R, I being ( ( ) ( complex real ext-real ) Real) st
( R : ( ( ) ( complex real ext-real ) Real) = Integral (M : ( ( Function-like V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) Element of K6(K7(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V51() ) set ) ) : ( ( ) ( non empty Relation-like V34() ) set ) ) : ( ( ) ( non empty ) set ) ) ,(Re f : ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) & I : ( ( ) ( complex real ext-real ) Real) = Integral (M : ( ( Function-like V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) Element of K6(K7(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V51() ) set ) ) : ( ( ) ( non empty Relation-like V34() ) set ) ) : ( ( ) ( non empty ) set ) ) ,(Im f : ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) & it : ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) = R : ( ( ) ( complex real ext-real ) Real) + (I : ( ( ) ( complex real ext-real ) Real) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) ) : ( ( ) ( complex ) set ) : ( ( ) ( complex ) set ) );
end;

theorem :: MESFUN6C:19
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V51() ) set ) -valued Function-like V34() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() 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 V90() V91() V92() 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 V51() ) set ) -valued Function-like V34() ) PartFunc of ,) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V34() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) = 0 : ( ( ) ( empty natural complex real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V45() V50() V51() V52() V53() V56() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V33() V34() V35() V36() V43() V44() V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V50() V51() V52() V53() V54() V55() V56() ) Element of K6(REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V34() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b5 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V34() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V51() ) set ) ) : ( ( ) ( non empty Relation-like V34() ) set ) ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUN6C:20
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,)
for E, A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) = 0 : ( ( ) ( empty natural complex real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V45() V50() V51() V52() V53() V56() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V33() V34() V35() V36() V43() V44() V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V50() V51() V52() V53() V54() V55() V56() ) Element of K6(REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b6 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUN6C:21
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) = 0 : ( ( ) ( empty natural complex real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V45() V50() V51() V52() V53() V56() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V33() V34() V35() V36() V43() V44() V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V50() V51() V52() V53() V54() V55() V56() ) Element of K6(REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b5 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b5 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( complex ) ( complex ) number ) = 0 : ( ( ) ( empty natural complex real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V45() V50() V51() V52() V53() V56() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V33() V34() V35() V36() V43() V44() V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V50() V51() V52() V53() V54() V55() V56() ) Element of K6(REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: MESFUN6C:22
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,)
for E, A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) = 0 : ( ( ) ( empty natural complex real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V45() V50() V51() V52() V53() V56() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V33() V34() V35() V36() V43() V44() V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V50() V51() V52() V53() V54() V55() V56() ) Element of K6(REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty ) set ) ) ) holds
Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) | (E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) M11(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b5 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) \ b6 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) M11(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( complex ) ( complex ) number ) = Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( complex ) ( complex ) number ) ;

theorem :: MESFUN6C:23
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b5 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUN6C:24
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,)
for A, B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) | (A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) M11(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b5 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) \/ b6 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) M11(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( complex ) ( complex ) number ) = (Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b5 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) )) : ( ( complex ) ( complex ) number ) + (Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) | B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b6 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) )) : ( ( complex ) ( complex ) number ) : ( ( ) ( complex ) set ) ;

theorem :: MESFUN6C:25
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,)
for B, A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( ) ( ) Element of K6(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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b6 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( complex ) ( complex ) number ) = (Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b6 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) )) : ( ( complex ) ( complex ) number ) + (Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) | B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b5 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) )) : ( ( complex ) ( complex ) number ) : ( ( ) ( complex ) set ) ) ;

definition
let k be ( ( real ) ( complex real ext-real ) number ) ;
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 V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) ;
func f to_power k -> ( ( Function-like ) ( Relation-like X : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(k : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) means :: MESFUN6C:def 4
( dom it : ( ( Function-like ) ( Relation-like k : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(k : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(X : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(k : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) = dom f : ( ( Function-like V27(X : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(k : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like X : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(k : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(X : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(k : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) Element of K6(K7(X : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(k : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V51() ) set ) ) : ( ( ) ( non empty Relation-like V34() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(X : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(k : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) & ( for x being ( ( ) ( ) Element of X : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(k : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) st x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) in dom it : ( ( Function-like ) ( Relation-like k : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(k : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(X : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(k : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) holds
it : ( ( Function-like ) ( Relation-like k : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(k : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) = (f : ( ( Function-like V27(X : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(k : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like X : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(k : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(X : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(k : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) Element of K6(K7(X : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) Element of K6(K6(k : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V51() ) set ) ) : ( ( ) ( non empty Relation-like V34() ) set ) ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) to_power k : ( ( non empty ) ( non empty ) set ) : ( ( real ) ( complex real ext-real ) set ) ) );
end;

registration
let X be ( ( non empty ) ( non empty ) set ) ;
cluster Relation-like X : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() nonnegative for ( ( ) ( ) Element of K6(K7(X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ;
end;

registration
let k be ( ( real non negative ) ( complex real ext-real non negative ) number ) ;
let X be ( ( non empty ) ( non empty ) set ) ;
let f be ( ( Function-like nonnegative ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() nonnegative ) PartFunc of ,) ;
cluster f : ( ( Function-like nonnegative ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() nonnegative ) Element of K6(K7(X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) to_power k : ( ( real non negative ) ( complex real ext-real non negative ) set ) : ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) -> Function-like nonnegative ;
end;

theorem :: MESFUN6C:26
for k being ( ( real ) ( complex real ext-real ) number )
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) is nonnegative & 0 : ( ( ) ( empty natural complex real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V45() V50() V51() V52() V53() V56() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V33() V34() V35() V36() V43() V44() V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V50() V51() V52() V53() V54() V55() V56() ) Element of K6(REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty ) set ) ) ) <= k : ( ( real ) ( complex real ext-real ) number ) holds
f : ( ( Function-like ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) to_power k : ( ( real ) ( complex real ext-real ) number ) : ( ( Function-like ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) is nonnegative ;

theorem :: MESFUN6C:27
for x being ( ( ) ( ) set )
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) is nonnegative holds
(f : ( ( Function-like ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) . x : ( ( ) ( ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) to_power (1 : ( ( ) ( non empty natural complex real ext-real positive non negative V43() V44() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( V50() V51() V52() V53() V54() V55() V56() ) Element of K6(REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty ) set ) ) ) / 2 : ( ( ) ( non empty natural complex real ext-real positive non negative V43() V44() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( V50() V51() V52() V53() V54() V55() V56() ) Element of K6(REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty complex real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) = sqrt (f : ( ( Function-like ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) . x : ( ( ) ( ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) ;

theorem :: MESFUN6C:28
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,)
for a being ( ( ) ( complex real ext-real ) Real) st A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) : ( ( ) ( ) Element of K6(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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ (less_dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) ,a : ( ( ) ( complex real ext-real ) Real) )) : ( ( ) ( ) set ) : ( ( ) ( ) Element of K6(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 V90() V91() V92() 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 V90() V91() V92() 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 REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) ,a : ( ( ) ( complex real ext-real ) Real) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUN6C:29
for k being ( ( real ) ( complex real ext-real ) number )
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) is nonnegative & 0 : ( ( ) ( empty natural complex real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V45() V50() V51() V52() V53() V56() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V33() V34() V35() V36() V43() V44() V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V50() V51() V52() V53() V54() V55() V56() ) Element of K6(REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty ) set ) ) ) <= k : ( ( real ) ( complex real ext-real ) number ) & E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) c= dom f : ( ( Function-like ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) : ( ( ) ( ) Element of K6(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) to_power k : ( ( real ) ( complex real ext-real ) number ) : ( ( Function-like ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUN6C:30
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) holds
|.f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) .| : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUN6C:31
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) st ex A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) holds
( f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) iff |.f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) .| : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ;

theorem :: MESFUN6C:32
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: MESFUN6C:33
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUN6C:34
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUN6C:35
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUN6C:36
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( complex ) ( complex ) number ) = (Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) | E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b6 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) )) : ( ( complex ) ( complex ) number ) + (Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) | E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b6 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) )) : ( ( complex ) ( complex ) number ) : ( ( ) ( complex ) set ) ) ;

theorem :: MESFUN6C:37
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) = (Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) | E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b6 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) + (Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((- g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) | E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b6 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) ) ;

theorem :: MESFUN6C:38
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,)
for r being ( ( ) ( complex real ext-real ) Real) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( r : ( ( ) ( complex real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(r : ( ( ) ( complex real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( complex ) ( complex ) number ) = r : ( ( ) ( complex real ext-real ) Real) * (Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) )) : ( ( complex ) ( complex ) number ) : ( ( ) ( complex ) set ) ) ;

theorem :: MESFUN6C:39
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(<i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( complex ) ( complex ) number ) = <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) * (Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) )) : ( ( complex ) ( complex ) number ) : ( ( ) ( complex ) set ) ) ;

theorem :: MESFUN6C:40
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,)
for c being ( ( complex ) ( complex ) number ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( c : ( ( complex ) ( complex ) number ) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(c : ( ( complex ) ( complex ) number ) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( complex ) ( complex ) number ) = c : ( ( complex ) ( complex ) number ) * (Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) )) : ( ( complex ) ( complex ) number ) : ( ( ) ( complex ) set ) ) ;

theorem :: MESFUN6C:41
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,)
for Y being ( ( ) ( ) set )
for r being ( ( ) ( complex real ext-real ) Real) holds (r : ( ( ) ( complex real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) | Y : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( ) ( ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) = r : ( ( ) ( complex real ext-real ) Real) (#) (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) | Y : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( ) ( ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUN6C:42
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) st ex A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) - f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) is nonnegative holds
ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) | E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b6 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) <= Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) | E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b6 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) ) ;

theorem :: MESFUN6C:43
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) st ex A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
|.(Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) )) : ( ( complex ) ( complex ) number ) .| : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) <= Integral (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,|.f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) .| : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) 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 V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ;
let B be ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
func Integral_on (M,B,f) -> ( ( complex ) ( complex ) number ) equals :: MESFUN6C:def 5
 Integral (M : ( ( Function-like nonnegative ) ( Relation-like S : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() nonnegative ) Element of K6(K7(S : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) ,(f : ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() V34() V35() ) set ) ) : ( ( ) ( non empty ) set ) ) | B : ( ( ) ( ) Element of S : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined B : ( ( ) ( ) Element of S : ( ( ) ( ) set ) ) -defined X : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() V34() V35() ) Element of K6(K7(X : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( complex ) ( complex ) number ) ;
end;

theorem :: MESFUN6C:44
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,)
for B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & Integral_on (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( complex ) ( complex ) number ) = (Integral_on (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) )) : ( ( complex ) ( complex ) number ) + (Integral_on (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) )) : ( ( complex ) ( complex ) number ) : ( ( ) ( complex ) set ) ) ;

theorem :: MESFUN6C:45
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,)
for c being ( ( complex ) ( complex ) number )
for B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
Integral_on (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(c : ( ( complex ) ( complex ) number ) (#) f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) ) : ( ( ) ( non empty Relation-like V33() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( complex ) ( complex ) number ) = c : ( ( complex ) ( complex ) number ) * (Integral_on (M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V51() ) set ) -valued Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined COMPLEX : ( ( ) ( non empty V45() V50() V56() ) set ) -valued Function-like V33() ) PartFunc of ,) )) : ( ( complex ) ( complex ) number ) : ( ( ) ( complex ) set ) ;

begin

theorem :: MESFUN6C:46
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,)
for a being ( ( ) ( complex real ext-real ) Real) st A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) : ( ( ) ( ) Element of K6(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 V90() V91() V92() 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 REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) ,a : ( ( ) ( complex real ext-real ) Real) )) : ( ( ) ( ) set ) : ( ( ) ( ) Element of K6(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 V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ (less_dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) ,a : ( ( ) ( complex real ext-real ) Real) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUN6C:47
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,)
for a being ( ( ) ( complex real ext-real ) Real) st A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) : ( ( ) ( ) Element of K6(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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ (great_dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) ,a : ( ( ) ( complex real ext-real ) Real) )) : ( ( ) ( ) set ) : ( ( ) ( ) Element of K6(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 V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ (less_eq_dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) ,a : ( ( ) ( complex real ext-real ) Real) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUN6C:48
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,)
for a being ( ( ) ( complex real ext-real ) Real) st A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) : ( ( ) ( ) Element of K6(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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ (less_eq_dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) ,a : ( ( ) ( complex real ext-real ) Real) )) : ( ( ) ( ) set ) : ( ( ) ( ) Element of K6(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 V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ (great_dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) ,a : ( ( ) ( complex real ext-real ) Real) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUN6C:49
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() 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 V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,)
for a being ( ( ) ( complex real ext-real ) Real) holds A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V90() V91() V92() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ (eq_dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) ,a : ( ( ) ( complex real ext-real ) Real) )) : ( ( ) ( ) set ) : ( ( ) ( ) Element of K6(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 V90() V91() V92() 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 REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) ,a : ( ( ) ( complex real ext-real ) Real) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) /\ (less_eq_dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V45() V50() V51() V52() V56() ) set ) -valued Function-like V33() V34() V35() ) PartFunc of ,) ,a : ( ( ) ( complex real ext-real ) Real) )) : ( ( ) ( ) set ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ;