:: MEASURE3 semantic presentation

begin

theorem :: MEASURE3:1
for F1, F2 being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) holds (Ser F1 : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) <= (Ser F2 : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) holds
SUM F1 : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) <= SUM F2 : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ;

theorem :: MEASURE3:2
for F1, F2 being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) holds (Ser F1 : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = (Ser F2 : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) holds
SUM F1 : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = SUM F2 : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ;

notation
let X be ( ( ) ( ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ;
synonym N_Sub_fam of S for N_Measure_fam of S;
end;

definition
let X be ( ( ) ( ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ;
let F be ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() 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 X : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() 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 X : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() 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 X : ( ( ) ( ) set ) ) ) ;
:: original: rng
redefine func rng F -> ( ( ) ( non empty V47() ) N_Measure_fam of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;
end;

theorem :: MEASURE3:3
for X 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 : ( ( ) ( ) set ) )
for M being ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) )
for F being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() 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 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) st meet (rng F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V47() ) N_Measure_fam of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) 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 : ( ( ) ( ) set ) ) ) c= F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) holds
M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) . (meet (rng F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V47() ) N_Measure_fam of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ;

theorem :: MEASURE3:4
for X 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 : ( ( ) ( ) set ) )
for G, F being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() 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 : ( ( ) ( ) set ) ) ) st G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) set ) & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
( G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) \ (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) & F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) c= F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ) holds
union (rng G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V47() ) N_Measure_fam of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) \ (meet (rng F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V47() ) N_Measure_fam of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ;

theorem :: MEASURE3:5
for X 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 : ( ( ) ( ) set ) )
for G, F being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() 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 : ( ( ) ( ) set ) ) ) st G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) set ) & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
( G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) \ (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) & F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) c= F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ) holds
meet (rng F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V47() ) N_Measure_fam of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) \ (union (rng G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V47() ) N_Measure_fam of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ;

theorem :: MEASURE3:6
for X 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 : ( ( ) ( ) set ) )
for M being ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) )
for G, F being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() 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 : ( ( ) ( ) set ) ) ) st M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) . (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) < +infty : ( ( ) ( V23() ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) & G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) set ) & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
( G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) \ (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) & F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) c= F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ) holds
M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) . (meet (rng F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V47() ) N_Measure_fam of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = (M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) . (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) - (M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) . (union (rng G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V47() ) N_Measure_fam of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ;

theorem :: MEASURE3:7
for X 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 : ( ( ) ( ) set ) )
for M being ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) )
for G, F being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() 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 : ( ( ) ( ) set ) ) ) st M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) . (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) < +infty : ( ( ) ( V23() ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) & G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) set ) & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
( G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) \ (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) & F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) c= F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ) holds
M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) . (union (rng G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V47() ) N_Measure_fam of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = (M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) . (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) - (M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) . (meet (rng F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V47() ) N_Measure_fam of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ;

theorem :: MEASURE3:8
for X 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 : ( ( ) ( ) set ) )
for M being ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) )
for G, F being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() 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 : ( ( ) ( ) set ) ) ) st M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) . (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) < +infty : ( ( ) ( V23() ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) & G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) set ) & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
( G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) \ (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) & F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) c= F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ) holds
M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) . (meet (rng F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V47() ) N_Measure_fam of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = (M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) . (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) - (sup (rng (M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) * G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty V47() ) ( non empty V47() V60() ) Element of bool ExtREAL : ( ( ) ( non empty V60() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ;

theorem :: MEASURE3:9
for X 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 : ( ( ) ( ) set ) )
for M being ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) )
for G, F being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() 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 : ( ( ) ( ) set ) ) ) st M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) . (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) < +infty : ( ( ) ( V23() ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) & G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) set ) & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
( G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) \ (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) & F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) c= F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ) holds
( M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) . (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) is ( ( ) ( V22() V23() ext-real ) Real) & inf (rng (M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) * F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty V47() ) ( non empty V47() V60() ) Element of bool ExtREAL : ( ( ) ( non empty V60() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) is ( ( ) ( V22() V23() ext-real ) Real) & sup (rng (M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) * G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty V47() ) ( non empty V47() V60() ) Element of bool ExtREAL : ( ( ) ( non empty V60() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) is ( ( ) ( V22() V23() ext-real ) Real) ) ;

theorem :: MEASURE3:10
for X 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 : ( ( ) ( ) set ) )
for M being ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) )
for G, F being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() 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 : ( ( ) ( ) set ) ) ) st M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) . (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) < +infty : ( ( ) ( V23() ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) & G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) set ) & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
( G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) \ (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) & F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) c= F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ) holds
sup (rng (M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) * G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty V47() ) ( non empty V47() V60() ) Element of bool ExtREAL : ( ( ) ( non empty V60() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = (M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) . (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) - (inf (rng (M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) * F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty V47() ) ( non empty V47() V60() ) Element of bool ExtREAL : ( ( ) ( non empty V60() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ;

theorem :: MEASURE3:11
for X 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 : ( ( ) ( ) set ) )
for M being ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) )
for G, F being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() 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 : ( ( ) ( ) set ) ) ) st M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) . (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) < +infty : ( ( ) ( V23() ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) & G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) set ) & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
( G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) \ (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) & F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) c= F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ) holds
inf (rng (M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) * F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty V47() ) ( non empty V47() V60() ) Element of bool ExtREAL : ( ( ) ( non empty V60() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = (M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) . (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) - (sup (rng (M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) * G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty V47() ) ( non empty V47() V60() ) Element of bool ExtREAL : ( ( ) ( non empty V60() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ;

theorem :: MEASURE3:12
for X 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 : ( ( ) ( ) set ) )
for M being ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) )
for F being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() 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 : ( ( ) ( ) set ) ) ) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) holds F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) c= F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) & M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) . (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) < +infty : ( ( ) ( V23() ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) holds
M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) . (meet (rng F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V47() ) N_Measure_fam of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = inf (rng (M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) * F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty V47() ) ( non empty V47() V60() ) Element of bool ExtREAL : ( ( ) ( non empty V60() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ;

theorem :: MEASURE3:13
for X 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 : ( ( ) ( ) set ) )
for M being ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative V90(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative V90(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) )
for F being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V53() ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V53() ) Sep_Sequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) holds SUM (M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative V90(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative V90(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) * F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V53() ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V53() ) Sep_Sequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) <= M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative V90(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative V90(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (union (rng F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V53() ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V53() ) Sep_Sequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V47() ) N_Measure_fam of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ;

theorem :: MEASURE3:14
for X 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 : ( ( ) ( ) set ) )
for M being ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative V90(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative V90(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) st ( for F being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V53() ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V53() ) Sep_Sequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) holds M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative V90(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative V90(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (union (rng F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V53() ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V53() ) Sep_Sequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V47() ) N_Measure_fam of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) <= SUM (M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative V90(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative V90(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) * F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V53() ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V53() ) Sep_Sequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) holds
M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative V90(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative V90(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) is ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ;

definition
let X be ( ( ) ( ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ;
let M be ( ( Function-like V30(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) V30(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ;
pred M is_complete S means :: MEASURE3:def 1
 for A being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) )
for B being ( ( ) ( ) set ) st B : ( ( ) ( ) set ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) c= B : ( ( ) ( ) set ) & M : ( ( ) ( non empty V47() ) N_Sub_fam of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . B : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = 0. : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) holds
A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ;
let M be ( ( Function-like V30(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) V30(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ;
mode thin of M -> ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) means :: MEASURE3:def 2
 ex B being ( ( ) ( ) set ) st
( B : ( ( ) ( ) set ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & it : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) c= B : ( ( ) ( ) set ) & M : ( ( ) ( non empty V47() ) N_Sub_fam of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . B : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = 0. : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) );
end;

definition
let X be ( ( ) ( ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ;
let M be ( ( Function-like V30(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) V30(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ;
func COM (S,M) -> ( ( non empty ) ( non empty ) Subset-Family of ) means :: MEASURE3:def 3
 for A being ( ( ) ( ) set ) holds
( A : ( ( ) ( ) set ) in it : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) iff ex B being ( ( ) ( ) set ) st
( B : ( ( ) ( ) set ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & ex C being ( ( ) ( ) thin of M : ( ( ) ( non empty V47() ) N_Sub_fam of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) st A : ( ( ) ( ) set ) = B : ( ( ) ( ) set ) \/ C : ( ( ) ( ) thin of M : ( ( Function-like V30(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) V30(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) set ) ) );
end;

definition
let X be ( ( ) ( ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ;
let M be ( ( Function-like V30(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) V30(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ;
let A be ( ( ) ( ) Element of COM (S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ,M : ( ( Function-like V30(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) V30(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ;
func MeasPart A -> ( ( non empty ) ( non empty ) Subset-Family of ) means :: MEASURE3:def 4
 for B being ( ( ) ( ) set ) holds
( B : ( ( ) ( ) set ) in it : ( ( Function-like V30(X : ( ( ) ( ) set ) ,M : ( ( ) ( non empty V47() ) N_Sub_fam of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ) ( V1() V4(X : ( ( ) ( ) set ) ) V5(M : ( ( ) ( non empty V47() ) N_Sub_fam of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function-like non empty V14(X : ( ( ) ( ) set ) ) V30(X : ( ( ) ( ) set ) ,M : ( ( ) ( non empty V47() ) N_Sub_fam of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) V67() ) Element of bool [:X : ( ( ) ( ) set ) ,M : ( ( ) ( non empty V47() ) N_Sub_fam of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) iff ( B : ( ( ) ( ) set ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & B : ( ( ) ( ) set ) c= A : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) & A : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) \ B : ( ( ) ( ) set ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) is ( ( ) ( ) thin of M : ( ( ) ( non empty V47() ) N_Sub_fam of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ) );
end;

theorem :: MEASURE3:15
for X 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 : ( ( ) ( ) set ) )
for M being ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) )
for F being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , COM (S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ex G being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() 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 : ( ( ) ( ) set ) ) ) st
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) holds G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) in MeasPart (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ;

theorem :: MEASURE3:16
for X 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 : ( ( ) ( ) set ) )
for M being ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) )
for F being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , COM (S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) )
for G being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() 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 : ( ( ) ( ) set ) ) ) ex H being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) st
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) holds H : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) \ (G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool (b4 : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) . b7 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MEASURE3:17
for X 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 : ( ( ) ( ) set ) )
for M being ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) )
for F being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) holds F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . n : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) is ( ( ) ( ) thin of M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) ) holds
ex G being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() 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 : ( ( ) ( ) set ) ) ) st
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
( F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) c= G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) & M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) . (G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V5(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V59() V60() V61() V62() V63() V64() V65() ) Element of bool REAL : ( ( ) ( non empty V33() V59() V60() V61() V65() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = 0. : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() V23() ext-real V59() V60() V61() V62() V63() V64() V65() ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ) ;

theorem :: MEASURE3:18
for X 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 : ( ( ) ( ) set ) )
for M being ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) )
for D being ( ( non empty ) ( non empty ) Subset-Family of ) st ( for A being ( ( ) ( ) set ) holds
( A : ( ( ) ( ) set ) in D : ( ( non empty ) ( non empty ) Subset-Family of ) iff ex B being ( ( ) ( ) set ) st
( B : ( ( ) ( ) set ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) & ex C being ( ( ) ( ) thin of M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) st A : ( ( ) ( ) set ) = B : ( ( ) ( ) set ) \/ C : ( ( ) ( ) thin of b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) set ) ) ) ) holds
D : ( ( non empty ) ( non empty ) Subset-Family of ) is ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ;

registration
let X be ( ( ) ( ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ;
let M be ( ( Function-like V30(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) V30(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ;
cluster COM (S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,M : ( ( Function-like V30(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) V30(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Element of bool (bool X : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( non empty V67() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) -> non empty compl-closed sigma-additive ;
end;

theorem :: MEASURE3:19
for X 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 : ( ( ) ( ) set ) )
for M being ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) )
for B1, B2 being ( ( ) ( ) set ) st B1 : ( ( ) ( ) set ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) & B2 : ( ( ) ( ) set ) in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) holds
for C1, C2 being ( ( ) ( ) thin of M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) st B1 : ( ( ) ( ) set ) \/ C1 : ( ( ) ( ) thin of b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) set ) = B2 : ( ( ) ( ) set ) \/ C2 : ( ( ) ( ) thin of b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) set ) holds
M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) . B1 : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) . B2 : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ;

definition
let X be ( ( ) ( ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ;
let M be ( ( Function-like V30(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) V30(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ;
func COM M -> ( ( Function-like V30( COM (S : ( ( ) ( ) set ) ,M : ( ( Function-like V30(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(S : ( ( ) ( ) set ) ) V30(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( V67() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4( COM (S : ( ( ) ( ) set ) ,M : ( ( Function-like V30(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(S : ( ( ) ( ) set ) ) V30(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( V67() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( COM (S : ( ( ) ( ) set ) ,M : ( ( Function-like V30(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(S : ( ( ) ( ) set ) ) V30(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( V67() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) V30( COM (S : ( ( ) ( ) set ) ,M : ( ( Function-like V30(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(S : ( ( ) ( ) set ) ) V30(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( V67() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of COM (S : ( ( ) ( ) set ) ,M : ( ( Function-like V30(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(S : ( ( ) ( ) set ) ) V30(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( V67() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) means :: MEASURE3:def 5
 for B being ( ( ) ( ) set ) st B : ( ( ) ( ) set ) in S : ( ( ) ( ) set ) holds
for C being ( ( ) ( ) thin of M : ( ( Function-like V30(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(S : ( ( ) ( ) set ) ) V30(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( V67() ) set ) : ( ( ) ( non empty ) set ) ) ) holds it : ( ( ) ( ) Element of S : ( ( ) ( ) set ) ) . (B : ( ( ) ( ) set ) \/ C : ( ( ) ( ) thin of M : ( ( Function-like V30(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) V30(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of X : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) = M : ( ( Function-like V30(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(S : ( ( ) ( ) set ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(S : ( ( ) ( ) set ) ) V30(S : ( ( ) ( ) set ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) Element of bool [:S : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V60() ) set ) :] : ( ( ) ( V67() ) set ) : ( ( ) ( non empty ) set ) ) . B : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V60() ) set ) ) ;
end;

theorem :: MEASURE3:20
for X 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 : ( ( ) ( ) set ) )
for M being ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) holds COM M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) : ( ( Function-like V30( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Subset-Family of ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Subset-Family of ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Subset-Family of ) ) V30( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Subset-Family of ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V67() V75() nonnegative sigma-additive ) sigma_Measure of COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,b3 : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Subset-Family of ) ) is_complete COM (S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ,M : ( ( Function-like V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V60() ) set ) ) V75() nonnegative sigma-additive ) ( V1() V4(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V5( ExtREAL : ( ( ) ( non empty V60() ) set ) ) Function-like non empty V14(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V30(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) SigmaField of b1 : ( ( ) ( ) 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 : ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V55() V56() V57() sigma-additive ) Subset-Family of ) ;