:: MESFUNC6 semantic presentation

begin

theorem :: MESFUNC6:1
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) holds |.(R_EAL f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) .| : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) = R_EAL (abs f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC6:2
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) PartFunc of ,)
for r being ( ( ) ( V24() real ext-real ) Real) st dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = r : ( ( ) ( V24() real ext-real ) Real) ) holds
f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: MESFUNC6:3
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for a being ( ( real ) ( V24() real ext-real ) number )
for x being ( ( ) ( ) set ) holds
( x : ( ( ) ( ) set ) in less_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,a : ( ( real ) ( V24() real ext-real ) number ) ) : ( ( ) ( ) set ) iff ( x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & ex y being ( ( ) ( V24() real ext-real ) Real) st
( y : ( ( ) ( V24() real ext-real ) Real) = f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) & y : ( ( ) ( V24() real ext-real ) Real) < a : ( ( real ) ( V24() real ext-real ) number ) ) ) ) ;

theorem :: MESFUNC6:4
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for a being ( ( real ) ( V24() real ext-real ) number )
for x being ( ( ) ( ) set ) holds
( x : ( ( ) ( ) set ) in less_eq_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,a : ( ( real ) ( V24() real ext-real ) number ) ) : ( ( ) ( ) set ) iff ( x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & ex y being ( ( ) ( V24() real ext-real ) Real) st
( y : ( ( ) ( V24() real ext-real ) Real) = f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) & y : ( ( ) ( V24() real ext-real ) Real) <= a : ( ( real ) ( V24() real ext-real ) number ) ) ) ) ;

theorem :: MESFUNC6:5
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for r being ( ( ) ( V24() real ext-real ) Real)
for x being ( ( ) ( ) set ) holds
( x : ( ( ) ( ) set ) in great_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,r : ( ( ) ( V24() real ext-real ) Real) ) : ( ( ) ( ) set ) iff ( x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & ex y being ( ( ) ( V24() real ext-real ) Real) st
( y : ( ( ) ( V24() real ext-real ) Real) = f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) & r : ( ( ) ( V24() real ext-real ) Real) < y : ( ( ) ( V24() real ext-real ) Real) ) ) ) ;

theorem :: MESFUNC6:6
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for r being ( ( ) ( V24() real ext-real ) Real)
for x being ( ( ) ( ) set ) holds
( x : ( ( ) ( ) set ) in great_eq_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,r : ( ( ) ( V24() real ext-real ) Real) ) : ( ( ) ( ) set ) iff ( x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & ex y being ( ( ) ( V24() real ext-real ) Real) st
( y : ( ( ) ( V24() real ext-real ) Real) = f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) & r : ( ( ) ( V24() real ext-real ) Real) <= y : ( ( ) ( V24() real ext-real ) Real) ) ) ) ;

theorem :: MESFUNC6:7
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for r being ( ( ) ( V24() real ext-real ) Real)
for x being ( ( ) ( ) set ) holds
( x : ( ( ) ( ) set ) in eq_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,r : ( ( ) ( V24() real ext-real ) Real) ) : ( ( ) ( ) set ) iff ( x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & ex y being ( ( ) ( V24() real ext-real ) Real) st
( y : ( ( ) ( V24() real ext-real ) Real) = f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) & r : ( ( ) ( V24() real ext-real ) Real) = y : ( ( ) ( V24() real ext-real ) Real) ) ) ) ;

theorem :: MESFUNC6:8
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( ) ( ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for F being ( ( Function-like V32( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ( V7() V10( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V11(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V32( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for r being ( ( ) ( V24() real ext-real ) Real) st ( for n being ( ( natural ) ( ordinal natural V24() real ext-real non negative ) Nat) holds F : ( ( Function-like V32( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ( V7() V10( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V11(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V32( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( ordinal natural V24() real ext-real non negative ) Nat) : ( ( ) ( ) set ) = Y : ( ( ) ( ) set ) /\ (great_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,(r : ( ( ) ( V24() real ext-real ) Real) - (1 : ( ( ) ( non empty ordinal natural V24() real ext-real positive non negative V29() V59() V60() V61() V62() V63() V64() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) / (n : ( ( natural ) ( ordinal natural V24() real ext-real non negative ) Nat) + 1 : ( ( ) ( non empty ordinal natural V24() real ext-real positive non negative V29() V59() V60() V61() V62() V63() V64() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V24() real ext-real positive non negative V29() V59() V60() V61() V62() V63() V64() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V24() real ext-real non negative rational ) Element of COMPLEX : ( ( ) ( non empty V35() V59() V65() ) set ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) )) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) holds
Y : ( ( ) ( ) set ) /\ (great_eq_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,r : ( ( ) ( V24() real ext-real ) Real) )) : ( ( ) ( ) set ) : ( ( ) ( ) set ) = meet (rng F : ( ( Function-like V32( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ( V7() V10( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V11(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V32( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC6:9
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( ) ( ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for F being ( ( Function-like V32( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ( V7() V10( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V11(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V32( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for r being ( ( ) ( V24() real ext-real ) Real) st ( for n being ( ( natural ) ( ordinal natural V24() real ext-real non negative ) Nat) holds F : ( ( Function-like V32( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ( V7() V10( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V11(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V32( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( ordinal natural V24() real ext-real non negative ) Nat) : ( ( ) ( ) set ) = Y : ( ( ) ( ) set ) /\ (less_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,(r : ( ( ) ( V24() real ext-real ) Real) + (1 : ( ( ) ( non empty ordinal natural V24() real ext-real positive non negative V29() V59() V60() V61() V62() V63() V64() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) / (n : ( ( natural ) ( ordinal natural V24() real ext-real non negative ) Nat) + 1 : ( ( ) ( non empty ordinal natural V24() real ext-real positive non negative V29() V59() V60() V61() V62() V63() V64() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V24() real ext-real positive non negative V29() V59() V60() V61() V62() V63() V64() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V24() real ext-real non negative rational ) Element of COMPLEX : ( ( ) ( non empty V35() V59() V65() ) set ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) )) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) holds
Y : ( ( ) ( ) set ) /\ (less_eq_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,r : ( ( ) ( V24() real ext-real ) Real) )) : ( ( ) ( ) set ) : ( ( ) ( ) set ) = meet (rng F : ( ( Function-like V32( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ( V7() V10( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V11(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V32( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC6:10
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( ) ( ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for F being ( ( Function-like V32( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ( V7() V10( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V11(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V32( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for r being ( ( ) ( V24() real ext-real ) Real) st ( for n being ( ( natural ) ( ordinal natural V24() real ext-real non negative ) Nat) holds F : ( ( Function-like V32( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ( V7() V10( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V11(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V32( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( ordinal natural V24() real ext-real non negative ) Nat) : ( ( ) ( ) set ) = Y : ( ( ) ( ) set ) /\ (less_eq_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,(r : ( ( ) ( V24() real ext-real ) Real) - (1 : ( ( ) ( non empty ordinal natural V24() real ext-real positive non negative V29() V59() V60() V61() V62() V63() V64() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) / (n : ( ( natural ) ( ordinal natural V24() real ext-real non negative ) Nat) + 1 : ( ( ) ( non empty ordinal natural V24() real ext-real positive non negative V29() V59() V60() V61() V62() V63() V64() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V24() real ext-real positive non negative V29() V59() V60() V61() V62() V63() V64() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V24() real ext-real non negative rational ) Element of COMPLEX : ( ( ) ( non empty V35() V59() V65() ) set ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) )) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) holds
Y : ( ( ) ( ) set ) /\ (less_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,r : ( ( ) ( V24() real ext-real ) Real) )) : ( ( ) ( ) set ) : ( ( ) ( ) set ) = union (rng F : ( ( Function-like V32( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ( V7() V10( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V11(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V32( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC6:11
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( ) ( ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for F being ( ( Function-like V32( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ( V7() V10( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V11(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V32( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for r being ( ( ) ( V24() real ext-real ) Real) st ( for n being ( ( natural ) ( ordinal natural V24() real ext-real non negative ) Nat) holds F : ( ( Function-like V32( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ( V7() V10( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V11(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V32( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( ordinal natural V24() real ext-real non negative ) Nat) : ( ( ) ( ) set ) = Y : ( ( ) ( ) set ) /\ (great_eq_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,(r : ( ( ) ( V24() real ext-real ) Real) + (1 : ( ( ) ( non empty ordinal natural V24() real ext-real positive non negative V29() V59() V60() V61() V62() V63() V64() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) / (n : ( ( natural ) ( ordinal natural V24() real ext-real non negative ) Nat) + 1 : ( ( ) ( non empty ordinal natural V24() real ext-real positive non negative V29() V59() V60() V61() V62() V63() V64() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V24() real ext-real positive non negative V29() V59() V60() V61() V62() V63() V64() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V24() real ext-real non negative rational ) Element of COMPLEX : ( ( ) ( non empty V35() V59() V65() ) set ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) )) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) holds
Y : ( ( ) ( ) set ) /\ (great_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,r : ( ( ) ( V24() real ext-real ) Real) )) : ( ( ) ( ) set ) : ( ( ) ( ) set ) = union (rng F : ( ( Function-like V32( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ( V7() V10( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V11(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V32( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) 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 V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let f be ( ( Function-like ) ( V7() V10(X : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ;
let A be ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
pred f is_measurable_on A means :: MESFUNC6:def 1
 R_EAL f : ( ( Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(X : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on A : ( ( Function-like ) ( V7() V10(X : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

theorem :: MESFUNC6:12
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) iff for r being ( ( real ) ( V24() real ext-real ) number ) holds A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ (less_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,r : ( ( real ) ( V24() real ext-real ) number ) )) : ( ( ) ( ) set ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:13
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) iff for r being ( ( real ) ( V24() real ext-real ) number ) holds A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ (great_eq_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,r : ( ( real ) ( V24() real ext-real ) number ) )) : ( ( ) ( ) set ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:14
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) iff for r being ( ( real ) ( V24() real ext-real ) number ) holds A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ (less_eq_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,r : ( ( real ) ( V24() real ext-real ) number ) )) : ( ( ) ( ) set ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:15
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) iff for r being ( ( real ) ( V24() real ext-real ) number ) holds A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ (great_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,r : ( ( real ) ( V24() real ext-real ) number ) )) : ( ( ) ( ) set ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:16
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for B, A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:17
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A, B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) M7(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )) ;

theorem :: MESFUNC6:18
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for r, s being ( ( ) ( V24() real ext-real ) Real) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
(A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ (great_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,r : ( ( ) ( V24() real ext-real ) Real) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (less_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,s : ( ( ) ( V24() real ext-real ) Real) )) : ( ( ) ( ) set ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: MESFUNC6:19
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f, g being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for r being ( ( ) ( V24() real ext-real ) Real) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
(A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ (less_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,r : ( ( ) ( V24() real ext-real ) Real) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (great_dom (g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,r : ( ( ) ( V24() real ext-real ) Real) )) : ( ( ) ( ) set ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: MESFUNC6:20
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for r being ( ( ) ( V24() real ext-real ) Real) holds R_EAL (r : ( ( ) ( V24() real ext-real ) Real) (#) f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) = r : ( ( ) ( V24() real ext-real ) Real) (#) (R_EAL f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC6:21
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for r being ( ( ) ( V24() real ext-real ) Real) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
r : ( ( ) ( V24() real ext-real ) Real) (#) f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

begin

theorem :: MESFUNC6:22
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) holds R_EAL f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) is V68() ;

theorem :: MESFUNC6:23
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) holds
( R_EAL (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) = (R_EAL f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) + (R_EAL g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) & R_EAL (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) - g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) = (R_EAL f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) - (R_EAL g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) & dom (R_EAL (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (dom (R_EAL f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom (R_EAL g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & dom (R_EAL (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) - g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (dom (R_EAL f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom (R_EAL g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & dom (R_EAL (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & dom (R_EAL (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) - g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:24
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f, g being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for r being ( ( ) ( V24() real ext-real ) Real)
for F being ( ( Function-like V32( RAT : ( ( ) ( non empty V35() V59() V60() V61() V62() V65() ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ( V7() V10( RAT : ( ( ) ( non empty V35() V59() V60() V61() V62() V65() ) set ) ) V11(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V32( RAT : ( ( ) ( non empty V35() V59() V60() V61() V62() V65() ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) Function of RAT : ( ( ) ( non empty V35() V59() V60() V61() V62() V65() ) set ) ,S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st ( for p being ( ( rational ) ( V24() real ext-real rational ) Rational) holds F : ( ( Function-like V32( RAT : ( ( ) ( non empty V35() V59() V60() V61() V62() V65() ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ( V7() V10( RAT : ( ( ) ( non empty V35() V59() V60() V61() V62() V65() ) set ) ) V11(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V32( RAT : ( ( ) ( non empty V35() V59() V60() V61() V62() V65() ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) Function of RAT : ( ( ) ( non empty V35() V59() V60() V61() V62() V65() ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . p : ( ( rational ) ( V24() real ext-real rational ) Rational) : ( ( ) ( ) set ) = (A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ (less_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,p : ( ( rational ) ( V24() real ext-real rational ) Rational) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ (less_dom (g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,(r : ( ( ) ( V24() real ext-real ) Real) - p : ( ( rational ) ( V24() real ext-real rational ) Rational) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) holds
A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ (less_dom ((f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ,r : ( ( ) ( V24() real ext-real ) Real) )) : ( ( ) ( ) set ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = union (rng F : ( ( Function-like V32( RAT : ( ( ) ( non empty V35() V59() V60() V61() V62() V65() ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ( V7() V10( RAT : ( ( ) ( non empty V35() V59() V60() V61() V62() V65() ) set ) ) V11(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V32( RAT : ( ( ) ( non empty V35() V59() V60() V61() V62() V65() ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) Function of RAT : ( ( ) ( non empty V35() V59() V60() V61() V62() V65() ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of bool b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ;

theorem :: MESFUNC6:25
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f, g being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for r being ( ( ) ( V24() real ext-real ) Real) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
ex F being ( ( Function-like V32( RAT : ( ( ) ( non empty V35() V59() V60() V61() V62() V65() ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ( V7() V10( RAT : ( ( ) ( non empty V35() V59() V60() V61() V62() V65() ) set ) ) V11(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V32( RAT : ( ( ) ( non empty V35() V59() V60() V61() V62() V65() ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) Function of RAT : ( ( ) ( non empty V35() V59() V60() V61() V62() V65() ) set ) ,S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
for p being ( ( rational ) ( V24() real ext-real rational ) Rational) holds F : ( ( Function-like V32( RAT : ( ( ) ( non empty V35() V59() V60() V61() V62() V65() ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ( V7() V10( RAT : ( ( ) ( non empty V35() V59() V60() V61() V62() V65() ) set ) ) V11(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V32( RAT : ( ( ) ( non empty V35() V59() V60() V61() V62() V65() ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) Function of RAT : ( ( ) ( non empty V35() V59() V60() V61() V62() V65() ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . p : ( ( rational ) ( V24() real ext-real rational ) Rational) : ( ( ) ( ) set ) = (A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ (less_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,p : ( ( rational ) ( V24() real ext-real rational ) Rational) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ (less_dom (g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,(r : ( ( ) ( V24() real ext-real ) Real) - p : ( ( rational ) ( V24() real ext-real rational ) Rational) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC6:26
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f, g being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:27
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) holds (R_EAL f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) - (R_EAL g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) = (R_EAL f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) + (R_EAL (- g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC6:28
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) holds
( - (R_EAL f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) = R_EAL ((- 1 : ( ( ) ( non empty ordinal natural V24() real ext-real positive non negative V29() V59() V60() V61() V62() V63() V64() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V24() real ext-real non positive rational ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) (#) f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) & - (R_EAL f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) = R_EAL (- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:29
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f, g being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) - g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

begin

theorem :: MESFUNC6:30
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) holds
( max+ (R_EAL f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) = max+ f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) & max- (R_EAL f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) = max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:31
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) holds 0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) <= (max+ f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) ;

theorem :: MESFUNC6:32
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) holds 0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) <= (max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) ;

theorem :: MESFUNC6:33
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) holds max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) = max+ (- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC6:34
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & 0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) < (max+ f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) holds
(max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) = 0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:35
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & 0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) < (max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) holds
(max+ f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) = 0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:36
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) holds
( dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = dom ((max+ f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) - (max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = dom ((max+ f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) + (max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:37
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
( ( (max+ f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) = f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) or (max+ f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) = 0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) & ( (max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) = - (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) or (max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) = 0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) ) ;

theorem :: MESFUNC6:38
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & (max+ f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) = f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) holds
(max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) = 0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:39
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & (max+ f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) = 0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
(max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) = - (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) ;

theorem :: MESFUNC6:40
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & (max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) = - (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) holds
(max+ f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) = 0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:41
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & (max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) = 0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
(max+ f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) = f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) ;

theorem :: MESFUNC6:42
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) holds f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) = (max+ f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) - (max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC6:43
for r being ( ( ) ( V24() real ext-real ) Real) holds |.r : ( ( ) ( V24() real ext-real ) Real) .| : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) = |.(R_EAL r : ( ( ) ( V24() real ext-real ) Real) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) .| : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ;

theorem :: MESFUNC6:44
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) holds R_EAL (abs f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) = |.(R_EAL f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) .| : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC6:45
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) holds abs f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) = (max+ f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) + (max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ;

begin

theorem :: MESFUNC6:46
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
max+ f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:47
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:48
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
abs f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let f be ( ( Function-like ) ( V7() V10(X : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ;
pred f is_simple_func_in S means :: MESFUNC6:def 2
 ex F being ( ( V120() ) ( V7() V10( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V11(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like FinSequence-like V120() ) Finite_Sep_Sequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) st
( dom f : ( ( Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool X : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = union (rng F : ( ( V120() ) ( V7() V10( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V11(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) Function-like FinSequence-like V120() ) Finite_Sep_Sequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of bool S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) & ( for n being ( ( natural ) ( ordinal natural V24() real ext-real non negative ) Nat)
for x, y being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st n : ( ( natural ) ( ordinal natural V24() real ext-real non negative ) Nat) in dom F : ( ( V120() ) ( V7() V10( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V11(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) Function-like FinSequence-like V120() ) Finite_Sep_Sequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V59() V60() V61() V62() V63() V64() ) Element of bool NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) & x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) in F : ( ( V120() ) ( V7() V10( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V11(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) Function-like FinSequence-like V120() ) Finite_Sep_Sequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( ordinal natural V24() real ext-real non negative ) Nat) : ( ( ) ( ) set ) & y : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) in F : ( ( V120() ) ( V7() V10( NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V11(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) Function-like FinSequence-like V120() ) Finite_Sep_Sequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( ordinal natural V24() real ext-real non negative ) Nat) : ( ( ) ( ) set ) holds
f : ( ( Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) = f : ( ( Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) . y : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) ) );
end;

theorem :: MESFUNC6:49
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) holds
( f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) iff R_EAL f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:50
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:51
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( ) ( ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) holds
( f : ( ( Function-like ) ( V7() V10(b1 : ( ( ) ( ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is nonnegative iff for x being ( ( ) ( ) set ) holds 0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) <= f : ( ( Function-like ) ( V7() V10(b1 : ( ( ) ( ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) ) ;

theorem :: MESFUNC6:52
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( ) ( ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) st ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( ) ( ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) holds
0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) <= f : ( ( Function-like ) ( V7() V10(b1 : ( ( ) ( ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) ) holds
f : ( ( Function-like ) ( V7() V10(b1 : ( ( ) ( ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is nonnegative ;

theorem :: MESFUNC6:53
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( ) ( ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) holds
( f : ( ( Function-like ) ( V7() V10(b1 : ( ( ) ( ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is nonpositive iff for x being ( ( ) ( ) set ) holds f : ( ( Function-like ) ( V7() V10(b1 : ( ( ) ( ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) <= 0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: MESFUNC6:54
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) st ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) <= 0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) holds
f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is nonpositive ;

theorem :: MESFUNC6:55
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( ) ( ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is nonnegative holds
f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | Y : ( ( ) ( ) set ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ;

theorem :: MESFUNC6:56
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is nonnegative & g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is nonnegative holds
f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ;

theorem :: MESFUNC6:57
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for r being ( ( ) ( V24() real ext-real ) Real) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is nonnegative holds
( ( 0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) <= r : ( ( ) ( V24() real ext-real ) Real) implies r : ( ( ) ( V24() real ext-real ) Real) (#) f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ) & ( r : ( ( ) ( V24() real ext-real ) Real) <= 0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) implies r : ( ( ) ( V24() real ext-real ) Real) (#) f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is nonpositive ) ) ;

theorem :: MESFUNC6:58
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) st ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in (dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) <= f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) ) holds
f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) - g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ;

theorem :: MESFUNC6:59
for X being ( ( non empty ) ( non empty ) set )
for f, g, h being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is nonnegative & g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is nonnegative & h : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is nonnegative holds
(f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) + h : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ;

theorem :: MESFUNC6:60
for X being ( ( non empty ) ( non empty ) set )
for f, g, h being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom ((f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) + h : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
((f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) + h : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) = ((f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) + (g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) + (h : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) ;

theorem :: MESFUNC6:61
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) holds
( max+ f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative & max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ) ;

theorem :: MESFUNC6:62
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) holds
( dom ((max+ (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) + (max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & dom ((max- (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) + (max+ f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & dom (((max+ (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) + (max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) + (max- g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & dom (((max- (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) + (max+ f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) + (max+ g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & (max+ (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) + (max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative & (max- (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) + (max+ f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ) ;

theorem :: MESFUNC6:63
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) holds ((max+ (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) + (max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) + (max- g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) = ((max- (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) + (max+ f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) + (max+ g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC6:64
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for r being ( ( ) ( V24() real ext-real ) Real) st 0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) <= r : ( ( ) ( V24() real ext-real ) Real) holds
( max+ (r : ( ( ) ( V24() real ext-real ) Real) (#) f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) = r : ( ( ) ( V24() real ext-real ) Real) (#) (max+ f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) & max- (r : ( ( ) ( V24() real ext-real ) Real) (#) f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) = r : ( ( ) ( V24() real ext-real ) Real) (#) (max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:65
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for r being ( ( ) ( V24() real ext-real ) Real) st 0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) <= r : ( ( ) ( V24() real ext-real ) Real) holds
( max+ ((- r : ( ( ) ( V24() real ext-real ) Real) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) (#) f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) = r : ( ( ) ( V24() real ext-real ) Real) (#) (max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) & max- ((- r : ( ( ) ( V24() real ext-real ) Real) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) (#) f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) = r : ( ( ) ( V24() real ext-real ) Real) (#) (max+ f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:66
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( ) ( ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) holds
( max+ (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | Y : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) = (max+ f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) | Y : ( ( ) ( ) set ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) & max- (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | Y : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) = (max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) | Y : ( ( ) ( ) set ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:67
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( ) ( ) set )
for f, g being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) st Y : ( ( ) ( ) set ) c= dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
( dom ((f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) | Y : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = Y : ( ( ) ( ) set ) & dom ((f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | Y : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) + (g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | Y : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = Y : ( ( ) ( ) set ) & (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) | Y : ( ( ) ( ) set ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) = (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | Y : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) + (g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | Y : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:68
for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for r being ( ( ) ( V24() real ext-real ) Real) holds eq_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,r : ( ( ) ( V24() real ext-real ) Real) ) : ( ( ) ( ) set ) = f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) " {r : ( ( ) ( V24() real ext-real ) Real) } : ( ( ) ( non empty V59() V60() V61() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

begin

theorem :: MESFUNC6:69
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for r, s being ( ( ) ( V24() real ext-real ) Real) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
(A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ (great_eq_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,r : ( ( ) ( V24() real ext-real ) Real) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (less_dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ,s : ( ( ) ( V24() real ext-real ) Real) )) : ( ( ) ( ) set ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: MESFUNC6:70
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: MESFUNC6:71
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) holds
dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:72
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f, g being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: MESFUNC6:73
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for r being ( ( ) ( V24() real ext-real ) Real) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) holds
r : ( ( ) ( V24() real ext-real ) Real) (#) f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: MESFUNC6:74
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) st ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) - g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) <= f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) ) holds
f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) - g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ;

theorem :: MESFUNC6:75
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for r being ( ( ) ( V24() real ext-real ) Real) ex f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) st
( f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) = r : ( ( ) ( V24() real ext-real ) Real) ) ) ;

theorem :: MESFUNC6:76
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for B, A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = (dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:77
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f, g being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
(max+ (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) + (max- f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:78
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f, g being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= (dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
(max- (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) + (max+ f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:79
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f, g being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) st dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & dom g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) holds
dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: MESFUNC6:80
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A, B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) iff f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) M7(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )) ) ;

theorem :: MESFUNC6:81
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) st ex A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = A : ( ( ) ( V24() real ext-real ) Real) holds
for c being ( ( ) ( V24() real ext-real ) Real)
for B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
c : ( ( ) ( V24() real ext-real ) Real) (#) f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let f be ( ( Function-like ) ( V7() V10(X : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ;
func Integral (M,f) -> ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) equals :: MESFUNC6:def 3
 Integral (M : ( ( Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ,(R_EAL f : ( ( Function-like ) ( V7() V10(X : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(X : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ;
end;

theorem :: MESFUNC6:82
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) st ex A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is nonnegative holds
Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(R_EAL f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ;

theorem :: MESFUNC6:83
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is nonnegative holds
( Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = integral+ (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(R_EAL f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) & Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = integral' (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(R_EAL f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) ;

theorem :: MESFUNC6:84
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) st ex A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is nonnegative holds
0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) <= Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ;

theorem :: MESFUNC6:85
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A, B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is nonnegative & A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | (A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) M7(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = (Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) + (Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ;

theorem :: MESFUNC6:86
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is nonnegative holds
0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) <= Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ;

theorem :: MESFUNC6:87
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A, B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is nonnegative & A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) <= Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ;

theorem :: MESFUNC6:88
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = 0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = 0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:89
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for E, A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = 0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | (E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) M7(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) 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 V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let f be ( ( Function-like ) ( V7() V10(X : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ;
pred f is_integrable_on M means :: MESFUNC6:def 4
 R_EAL f : ( ( Function-like ) ( V7() V10(X : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( V7() V10(X : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

theorem :: MESFUNC6:90
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( -infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) < Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) & Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) < +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) ;

theorem :: MESFUNC6:91
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:92
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for A, B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | (A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) M7(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = (Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) + (Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ;

theorem :: MESFUNC6:93
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for B, A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = (dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) \ A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = (Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) + (Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) ;

theorem :: MESFUNC6:94
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) st ex A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) holds
( f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) iff abs f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ;

theorem :: MESFUNC6:95
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
|.(Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) .| : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) <= Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(abs f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ;

theorem :: MESFUNC6:96
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) st ex A being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( A : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) = dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_measurable_on A : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) & dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = dom g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
abs (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) <= g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) ) holds
( f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(abs f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) <= Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) ;

theorem :: MESFUNC6:97
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for r being ( ( ) ( V24() real ext-real ) Real) st dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & 0 : ( ( ) ( empty ordinal natural V24() real ext-real non positive non negative V29() V59() V60() V61() V62() V63() V64() V65() rational ) Element of NAT : ( ( ) ( V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) <= r : ( ( ) ( V24() real ext-real ) Real) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) = r : ( ( ) ( V24() real ext-real ) Real) ) holds
Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = (R_EAL r : ( ( ) ( V24() real ext-real ) Real) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) * (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ;

theorem :: MESFUNC6:98
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is nonnegative & g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is nonnegative holds
f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:99
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: MESFUNC6:100
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC6:101
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = (dom f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = (Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) + (Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) ;

theorem :: MESFUNC6:102
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for r being ( ( ) ( V24() real ext-real ) Real) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( r : ( ( ) ( V24() real ext-real ) Real) (#) f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(r : ( ( ) ( V24() real ext-real ) Real) (#) f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = (R_EAL r : ( ( ) ( V24() real ext-real ) Real) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) * (Integral (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) 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 V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
let f be ( ( Function-like ) ( V7() V10(X : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ;
let B be ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
func Integral_on (M,B,f) -> ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) equals :: MESFUNC6:def 5
 Integral (M : ( ( Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) Element of bool [:S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ,(f : ( ( Function-like ) ( V7() V10(X : ( ( non empty ) ( non empty ) set ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V67() ) Element of bool [:X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) | B : ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( V7() V10(X : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:X : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ;
end;

theorem :: MESFUNC6:103
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom (f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & Integral_on (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) + g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = (Integral_on (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) + (Integral_on (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,g : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) ;

theorem :: MESFUNC6:104
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,)
for r being ( ( ) ( V24() real ext-real ) Real)
for B being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) | B : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & Integral_on (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(r : ( ( ) ( V24() real ext-real ) Real) (#) f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) ) : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) :] : ( ( ) ( non empty V66() V67() V68() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = (R_EAL r : ( ( ) ( V24() real ext-real ) Real) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) * (Integral_on (M : ( ( Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V7() V10(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V11( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like V32(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() 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 V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( V7() V10(b1 : ( ( non empty ) ( non empty ) set ) ) V11( REAL : ( ( ) ( non empty V35() V59() V60() V61() V65() ) set ) ) Function-like V66() V67() V68() ) PartFunc of ,) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) ;