:: PROB_4 semantic presentation

begin

definition
let X be ( ( ) ( ) set ) ;
let Si be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ;
let XSeq be ( ( V17(Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) V17( bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ) ;
let n be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ;
:: original: .
redefine func XSeq . n -> ( ( ) ( ) Element of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ;
end;

theorem :: PROB_4:1
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) )
for XSeq being ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) holds rng XSeq : ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) c= Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ;

theorem :: PROB_4:2
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) )
for f being ( ( V13() Function-like ) ( V13() Function-like ) Function) holds
( f : ( ( V13() Function-like ) ( V13() Function-like ) Function) is ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) iff f : ( ( V13() Function-like ) ( V13() Function-like ) Function) is ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ,b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ,Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ;

scheme :: PROB_4:sch 1
LambdaSigmaSSeq{ F1() -> ( ( ) ( ) set ) , F2() -> ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) , F3( ( ( ) ( ) set ) ) -> ( ( ) ( ) Element of F2( ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) } :
ex f being ( ( V17(F2( ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(F2( ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) V17( bool F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of F2( ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) st
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) holds f : ( ( V17(F2( ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(F2( ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) V17( bool F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of F2( ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) : ( ( ) ( ) Element of F2( ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) = F3( ( ( ) ( ) Element of F2( ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( ) ( ) Element of F2( ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of F1( ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )
proof end;

registration
let X be ( ( ) ( ) set ) ;
cluster non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V118() for ( ( ) ( ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ,(bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) :] : ( ( ) ( non empty V13() ) set ) : ( ( ) ( non empty V79() ) set ) ) ;
end;

registration
let X be ( ( ) ( ) set ) ;
let Si be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) ;
cluster non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V118() for ( ( ) ( ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ,(bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) :] : ( ( ) ( non empty V13() ) set ) : ( ( ) ( non empty V79() ) set ) ) ;
end;

theorem :: PROB_4:3
for X being ( ( ) ( ) set )
for A, B being ( ( ) ( ) Subset of ( ( ) ( non empty V79() ) set ) ) st A : ( ( ) ( ) Subset of ( ( ) ( non empty V79() ) set ) ) misses B : ( ( ) ( ) Subset of ( ( ) ( non empty V79() ) set ) ) holds
(A : ( ( ) ( ) Subset of ( ( ) ( non empty V79() ) set ) ) ,B : ( ( ) ( ) Subset of ( ( ) ( non empty V79() ) set ) ) ) followed_by {} : ( ( ) ( ) set ) : ( ( ) ( V13() Function-like ) set ) is disjoint_valued ;

theorem :: PROB_4:4
for X being ( ( ) ( ) set )
for S being ( ( non empty ) ( non empty ) set ) holds
( S : ( ( non empty ) ( non empty ) set ) is ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) ) iff ( S : ( ( non empty ) ( non empty ) set ) c= bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V79() ) set ) & ( for A1 being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) st rng A1 : ( ( ) ( ) Subset of ( ( ) ( non empty V79() ) set ) ) : ( ( non empty ) ( non empty ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) c= S : ( ( non empty ) ( non empty ) set ) holds
Union A1 : ( ( ) ( ) Subset of ( ( ) ( non empty V79() ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V79() ) set ) ) in S : ( ( non empty ) ( non empty ) set ) ) & ( for A being ( ( ) ( ) Subset of ( ( ) ( non empty V79() ) set ) ) st A : ( ( ) ( ) Subset of ( ( ) ( non empty V79() ) set ) ) in S : ( ( non empty ) ( non empty ) set ) holds
A : ( ( ) ( ) Subset of ( ( ) ( non empty V79() ) set ) ) ` : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V79() ) set ) ) in S : ( ( non empty ) ( non empty ) set ) ) ) ) ;

theorem :: PROB_4:5
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for A, B being ( ( ) ( ) Event of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (A : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) \ B : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) = (P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (A : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) \/ B : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) - (P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . B : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) ;

theorem :: PROB_4:6
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for A, B being ( ( ) ( ) Event of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st A : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= B : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . B : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) holds
P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . A : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ;

theorem :: PROB_4:7
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for ASeq being ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) holds P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (ASeq : ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) iff P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (Union ASeq : ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V79() ) set ) ) : ( ( ) ( V11() V12() ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ;

theorem :: PROB_4:8
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for ASeq being ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( ( for A being ( ( ) ( ) set ) st A : ( ( ) ( ) set ) in rng ASeq : ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) holds
P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . A : ( ( ) ( ) set ) : ( ( ) ( V11() V12() ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) iff P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (union (rng ASeq : ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V79() ) set ) ) : ( ( ) ( V11() V12() ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ;

theorem :: PROB_4:9
for seq being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) )
for Eseq being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) st seq : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) = Eseq : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) holds
Partial_Sums seq : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ,REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) :] : ( ( ) ( non empty V13() V33() V34() V35() ) set ) : ( ( ) ( non empty V79() ) set ) ) = Ser Eseq : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ,ExtREAL : ( ( ) ( non empty V51() ) set ) :] : ( ( ) ( non empty V13() V34() ) set ) : ( ( ) ( non empty V79() ) set ) ) ;

theorem :: PROB_4:10
for seq being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) )
for Eseq being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) st seq : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) = Eseq : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) & seq : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) is V101() holds
upper_bound seq : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) = sup (rng Eseq : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) ) : ( ( ) ( non empty V51() ) Element of bool ExtREAL : ( ( ) ( non empty V51() ) set ) : ( ( ) ( non empty V79() ) set ) ) : ( ( ext-real ) ( ext-real ) set ) ;

theorem :: PROB_4:11
for seq being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) )
for Eseq being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) st seq : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) = Eseq : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) & seq : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) is V102() holds
lower_bound seq : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) = inf (rng Eseq : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) ) : ( ( ) ( non empty V51() ) Element of bool ExtREAL : ( ( ) ( non empty V51() ) set ) : ( ( ) ( non empty V79() ) set ) ) : ( ( ext-real ) ( ext-real ) set ) ;

theorem :: PROB_4:12
for seq being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) )
for Eseq being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) st seq : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) = Eseq : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) & seq : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) is nonnegative & seq : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) is summable holds
Sum seq : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) = SUM Eseq : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) ;

theorem :: PROB_4:13
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) is ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

definition
let Omega be ( ( non empty ) ( non empty ) set ) ;
let Sigma be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ;
let P be ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) ;
func P2M P -> ( ( Function-like V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) equals :: PROB_4:def 1
P : ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ;
end;

theorem :: PROB_4:14
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . X : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ext-real ) set ) = R_EAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real positive non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) holds
M : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) is ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ;
let M be ( ( Function-like V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( non empty V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) ;
assume M : ( ( Function-like V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( non empty V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) ) ) . X : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ext-real ) set ) = R_EAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real positive non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V51() ) set ) ) ;
func M2P M -> ( ( ) ( non empty V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) equals :: PROB_4:def 2
M : ( ( ) ( non empty V13() V16(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27(S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ;
end;

theorem :: PROB_4:15
for X being ( ( ) ( ) set )
for A1 being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) st A1 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) is non-descending holds
Partial_Union A1 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) :] : ( ( ) ( non empty V13() ) set ) : ( ( ) ( non empty V79() ) set ) ) = A1 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ;

theorem :: PROB_4:16
for X being ( ( ) ( ) set )
for A1 being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) st A1 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) is non-descending holds
( (Partial_Diff_Union A1 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) disjoint_valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) :] : ( ( ) ( non empty V13() ) set ) : ( ( ) ( non empty V79() ) set ) ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) = A1 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) holds (Partial_Diff_Union A1 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) disjoint_valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) :] : ( ( ) ( non empty V13() ) set ) : ( ( ) ( non empty V79() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real positive non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real positive non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) = (A1 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real positive non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real positive non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) \ (A1 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ) ;

theorem :: PROB_4:17
for X being ( ( ) ( ) set )
for A1 being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) st A1 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) is non-descending holds
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) holds A1 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real positive non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real positive non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) = ((Partial_Diff_Union A1 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) disjoint_valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) :] : ( ( ) ( non empty V13() ) set ) : ( ( ) ( non empty V79() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real positive non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real positive non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) \/ (A1 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ;

theorem :: PROB_4:18
for X being ( ( ) ( ) set )
for A1 being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) st A1 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) is non-descending holds
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) holds (Partial_Diff_Union A1 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) disjoint_valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) :] : ( ( ) ( non empty V13() ) set ) : ( ( ) ( non empty V79() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real positive non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real positive non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) misses A1 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ;

theorem :: PROB_4:19
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) )
for XSeq being ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) st XSeq : ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) is non-descending holds
Partial_Union XSeq : ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) :] : ( ( ) ( non empty V13() ) set ) : ( ( ) ( non empty V79() ) set ) ) = XSeq : ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ;

theorem :: PROB_4:20
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) )
for XSeq being ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) st XSeq : ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) is non-descending holds
( (Partial_Diff_Union XSeq : ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) disjoint_valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) :] : ( ( ) ( non empty V13() ) set ) : ( ( ) ( non empty V79() ) set ) ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = XSeq : ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) holds (Partial_Diff_Union XSeq : ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) disjoint_valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) :] : ( ( ) ( non empty V13() ) set ) : ( ( ) ( non empty V79() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real positive non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real positive non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = (XSeq : ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real positive non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real positive non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) \ (XSeq : ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) ) ;

theorem :: PROB_4:21
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of X : ( ( ) ( ) set ) )
for XSeq being ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) st XSeq : ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) is non-descending holds
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) holds (Partial_Diff_Union XSeq : ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) disjoint_valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ,(bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) :] : ( ( ) ( non empty V13() ) set ) : ( ( ) ( non empty V79() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real positive non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real positive non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) misses XSeq : ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) V17( bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ;

definition
let Omega be ( ( non empty ) ( non empty ) set ) ;
let Sigma be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ;
let P be ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) ;
pred P is_complete Sigma means :: PROB_4:def 3
 for A being ( ( ) ( ) Subset of ( ( ) ( non empty V79() ) set ) )
for B being ( ( ) ( ) set ) st B : ( ( ) ( ) set ) in Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) & A : ( ( ) ( ) Subset of ( ( ) ( non empty V79() ) set ) ) c= B : ( ( ) ( ) set ) & P : ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) . B : ( ( ) ( ) set ) : ( ( ) ( V11() V12() ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) holds
A : ( ( ) ( ) Subset of ( ( ) ( non empty V79() ) set ) ) in Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ;
end;

theorem :: PROB_4:22
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) is_complete Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) iff P2M P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) is_complete Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

definition
let Omega be ( ( non empty ) ( non empty ) set ) ;
let Sigma be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ;
let P be ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) ;
mode thin of P -> ( ( ) ( ) Subset of ( ( ) ( non empty V79() ) set ) ) means :: PROB_4:def 4
 ex A being ( ( ) ( ) set ) st
( A : ( ( ) ( ) set ) in Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) & it : ( ( ) ( ) Element of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) c= A : ( ( ) ( ) set ) & P : ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) . A : ( ( ) ( ) set ) : ( ( ) ( V11() V12() ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) );
end;

theorem :: PROB_4:23
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for Y being ( ( ) ( ) Subset of ( ( ) ( non empty V79() ) set ) ) holds
( Y : ( ( ) ( ) Subset of ( ( ) ( non empty V79() ) set ) ) is ( ( ) ( ) thin of P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) iff Y : ( ( ) ( ) Subset of ( ( ) ( non empty V79() ) set ) ) is ( ( ) ( ) thin of P2M P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ) ;

theorem :: PROB_4:24
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds {} : ( ( ) ( ) set ) is ( ( ) ( ) thin of P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ;

theorem :: PROB_4:25
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for B1, B2 being ( ( ) ( ) set ) st B1 : ( ( ) ( ) set ) in Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & B2 : ( ( ) ( ) set ) in Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) holds
for C1, C2 being ( ( ) ( ) thin of P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) st B1 : ( ( ) ( ) set ) \/ C1 : ( ( ) ( ) thin of b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) set ) = B2 : ( ( ) ( ) set ) \/ C2 : ( ( ) ( ) thin of b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) set ) holds
P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . B1 : ( ( ) ( ) set ) : ( ( ) ( V11() V12() ext-real ) set ) = P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . B2 : ( ( ) ( ) set ) : ( ( ) ( V11() V12() ext-real ) set ) ;

definition
let Omega be ( ( non empty ) ( non empty ) set ) ;
let Sigma be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ;
let P be ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) ;
func COM (Sigma,P) -> ( ( non empty ) ( non empty ) Subset-Family of ) means :: PROB_4:def 5
 for A being ( ( ) ( ) set ) holds
( A : ( ( ) ( ) set ) in it : ( ( ) ( ) Element of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) iff ex B being ( ( ) ( ) set ) st
( B : ( ( ) ( ) set ) in Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) & ex C being ( ( ) ( ) thin of P : ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) st A : ( ( ) ( ) set ) = B : ( ( ) ( ) set ) \/ C : ( ( ) ( ) thin of P : ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) set ) ) );
end;

theorem :: PROB_4:26
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for C being ( ( ) ( ) thin of P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) holds C : ( ( ) ( ) thin of b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) in COM (Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ;

theorem :: PROB_4:27
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds COM (Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) = COM (Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,(P2M P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ;

definition
let Omega be ( ( non empty ) ( non empty ) set ) ;
let Sigma be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ;
let P be ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) ;
let A be ( ( ) ( ) Element of COM (Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ,P : ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ;
func P_COM2M_COM A -> ( ( ) ( ) Element of COM (Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ,(P2M P : ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( Function-like V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) equals :: PROB_4:def 6
A : ( ( ) ( ) Element of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ;
end;

theorem :: PROB_4:28
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) c= COM (Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ;

definition
let Omega be ( ( non empty ) ( non empty ) set ) ;
let Sigma be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ;
let P be ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) ;
let A be ( ( ) ( ) Element of COM (Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ,P : ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ;
func ProbPart A -> ( ( non empty ) ( non empty ) Subset-Family of ) means :: PROB_4:def 7
 for B being ( ( ) ( ) set ) holds
( B : ( ( ) ( ) set ) in it : ( ( ) ( ) Element of Omega : ( ( ) ( ) set ) ) iff ( B : ( ( ) ( ) set ) in Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) & B : ( ( ) ( ) set ) c= A : ( ( ) ( ) Element of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) & A : ( ( ) ( ) Element of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) \ B : ( ( ) ( ) set ) : ( ( ) ( ) Element of bool Omega : ( ( ) ( ) set ) : ( ( ) ( non empty V79() ) set ) ) is ( ( ) ( ) thin of P : ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ) );
end;

theorem :: PROB_4:29
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for A being ( ( ) ( ) Element of COM (Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) holds ProbPart A : ( ( ) ( ) Element of COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) = MeasPart (P_COM2M_COM A : ( ( ) ( ) Element of COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) : ( ( ) ( ) Element of COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,(P2M b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) : ( ( non empty ) ( non empty ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ;

theorem :: PROB_4:30
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for A being ( ( ) ( ) Element of COM (Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) )
for A1, A2 being ( ( ) ( ) set ) st A1 : ( ( ) ( ) set ) in ProbPart A : ( ( ) ( ) Element of COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) & A2 : ( ( ) ( ) set ) in ProbPart A : ( ( ) ( ) Element of COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) holds
P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . A1 : ( ( ) ( ) set ) : ( ( ) ( V11() V12() ext-real ) set ) = P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . A2 : ( ( ) ( ) set ) : ( ( ) ( V11() V12() ext-real ) set ) ;

theorem :: PROB_4:31
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for F being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , COM (Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ex BSeq being ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) holds BSeq : ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) in ProbPart (F : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( ) ( ) Element of COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ;

theorem :: PROB_4:32
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for F being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , COM (Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) )
for BSeq being ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ex CSeq being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) st
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) holds CSeq : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) = (F : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( ) ( ) Element of COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) \ (BSeq : ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool (b4 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) . b7 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( ) ( ) Element of COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) : ( ( ) ( non empty V79() ) set ) ) ;

theorem :: PROB_4:33
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for BSeq being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) holds BSeq : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) . n : ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) is ( ( ) ( ) thin of P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ) holds
ex CSeq being ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) holds
( BSeq : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) c= CSeq : ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (CSeq : ( ( V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) Function-like V23( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ;

theorem :: PROB_4:34
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) 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 Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & ex C being ( ( ) ( ) thin of P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) st A : ( ( ) ( ) set ) = B : ( ( ) ( ) set ) \/ C : ( ( ) ( ) thin of b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) set ) ) ) ) holds
D : ( ( non empty ) ( non empty ) Subset-Family of ) is ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ;

registration
let Omega be ( ( non empty ) ( non empty ) set ) ;
let Sigma be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ;
let P be ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) ;
cluster COM (Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ,P : ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) -> non empty compl-closed sigma-multiplicative ;
end;

definition
let Omega be ( ( non empty ) ( non empty ) set ) ;
let Sigma be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ;
let P be ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) ;
:: original: thin
redefine mode thin of P -> ( ( ) ( ) Event of COM (Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ,P : ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) ;
end;

theorem :: PROB_4:35
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for A being ( ( ) ( ) set ) holds
( A : ( ( ) ( ) set ) in COM (Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) iff ex A1, A2 being ( ( ) ( ) set ) st
( A1 : ( ( ) ( ) set ) in Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & A2 : ( ( ) ( ) set ) in Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & A1 : ( ( ) ( ) set ) c= A : ( ( ) ( ) set ) & A : ( ( ) ( ) set ) c= A2 : ( ( ) ( ) set ) & P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (A2 : ( ( ) ( ) set ) \ A1 : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of bool b6 : ( ( ) ( ) set ) : ( ( ) ( non empty V79() ) set ) ) : ( ( ) ( V11() V12() ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ) ;

theorem :: PROB_4:36
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for C being ( ( non empty ) ( non empty ) Subset-Family of ) st ( for A being ( ( ) ( ) set ) holds
( A : ( ( ) ( ) set ) in C : ( ( non empty ) ( non empty ) Subset-Family of ) iff ex A1, A2 being ( ( ) ( ) set ) st
( A1 : ( ( ) ( ) set ) in Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & A2 : ( ( ) ( ) set ) in Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & A1 : ( ( ) ( ) set ) c= A : ( ( ) ( ) set ) & A : ( ( ) ( ) set ) c= A2 : ( ( ) ( ) set ) & P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (A2 : ( ( ) ( ) set ) \ A1 : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of bool b7 : ( ( ) ( ) set ) : ( ( ) ( non empty V79() ) set ) ) : ( ( ) ( V11() V12() ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) ) ) holds
C : ( ( non empty ) ( non empty ) Subset-Family of ) = COM (Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) ;

definition
let Omega be ( ( non empty ) ( non empty ) set ) ;
let Sigma be ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ;
let P be ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) ;
func COM P -> ( ( ) ( non empty V13() V16( COM (Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ,P : ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23( COM (Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ,P : ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) V27( COM (Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ,P : ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of COM (Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ,P : ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) ) means :: PROB_4:def 8
 for B being ( ( ) ( ) set ) st B : ( ( ) ( ) set ) in Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) holds
for C being ( ( ) ( ) thin of P : ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ) holds it : ( ( ) ( ) Element of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) . (B : ( ( ) ( ) set ) \/ C : ( ( ) ( ) thin of P : ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) = P : ( ( ) ( non empty V13() V16(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) . B : ( ( ) ( ) set ) : ( ( ) ( V11() V12() ext-real ) set ) ;
end;

theorem :: PROB_4:37
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds COM P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty V13() V16( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) ) V27( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) ) = COM (P2M P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V27( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,(P2M b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( non empty V13() V16( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,(P2M b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,(P2M b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) V27( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,(P2M b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) Element of bool [:(COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,(P2M b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V42() nonnegative sigma-additive ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( ExtREAL : ( ( ) ( non empty V51() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V51() ) set ) ) V34() V42() nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) )) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Element of bool (bool b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V79() ) set ) : ( ( ) ( non empty V79() ) set ) ) ,ExtREAL : ( ( ) ( non empty V51() ) set ) :] : ( ( ) ( non empty V13() V34() ) set ) : ( ( ) ( non empty V79() ) set ) ) ;

theorem :: PROB_4:38
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds COM P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty V13() V16( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) ) V27( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) ) is_complete COM (Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) ;

theorem :: PROB_4:39
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for A being ( ( ) ( ) Event of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . A : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) = (COM P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty V13() V16( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) ) V27( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) ) . A : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) set ) ;

theorem :: PROB_4:40
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for C being ( ( ) ( ) thin of P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) holds (COM P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty V13() V16( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) ) V27( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) ) . C : ( ( ) ( ) thin of b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() V12() ext-real non negative V43() V49() V50() V51() V52() V53() V54() V55() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V50() V51() V52() V53() V54() V55() V56() ) Element of bool REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) : ( ( ) ( non empty V79() ) set ) ) ) ;

theorem :: PROB_4:41
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for A being ( ( ) ( ) Element of COM (Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) )
for B being ( ( ) ( ) set ) st B : ( ( ) ( ) set ) in ProbPart A : ( ( ) ( ) Element of COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) ) : ( ( non empty ) ( non empty ) Subset-Family of ) holds
P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . B : ( ( ) ( ) set ) : ( ( ) ( V11() V12() ext-real ) set ) = (COM P : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty V13() V16( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) ) V27( COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) ) . A : ( ( ) ( ) Element of COM (b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ,b3 : ( ( ) ( non empty V13() V16(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V17( REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) Function-like V23(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V27(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) V33() V34() V35() V42() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( non empty ) ( non empty compl-closed sigma-multiplicative V78() V79() V80() sigma-additive ) Subset-Family of ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V44() V50() V51() V52() V56() ) set ) ) ;