:: PROB_3 semantic presentation

begin

theorem :: PROB_3:1
for f being ( ( Relation-like Function-like FinSequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence) holds not 0 : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) in dom f : ( ( Relation-like Function-like FinSequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence) : ( ( ) ( V78() V79() V80() V81() V82() V83() ) Element of K10(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) ;

theorem :: PROB_3:2
for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat)
for f being ( ( Relation-like Function-like FinSequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence) holds
( n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) in dom f : ( ( Relation-like Function-like FinSequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence) : ( ( ) ( V78() V79() V80() V81() V82() V83() ) Element of K10(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) iff ( n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) <> 0 : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) & n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) <= len f : ( ( Relation-like Function-like FinSequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence) : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ) ;

theorem :: PROB_3:3
for g being ( ( real ) ( ext-real V39() real ) number )
for f being ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Real_Sequence) st ex k being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) st
for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) st k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) <= n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) holds
f : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Real_Sequence) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ext-real V39() real ) set ) = g : ( ( real ) ( ext-real V39() real ) number ) holds
( f : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Real_Sequence) is convergent & lim f : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Real_Sequence) : ( ( ) ( ext-real V39() real ) Element of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) = g : ( ( real ) ( ext-real V39() real ) number ) ) ;

theorem :: PROB_3:4
for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat)
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for ASeq being ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -valued bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) )
for P being ( ( ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V21(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) holds (P : ( ( ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V21(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) * ASeq : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -valued bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ext-real V39() real ) set ) >= 0 : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ;

theorem :: PROB_3:5
for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat)
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for ASeq, BSeq being ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -valued bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) )
for P being ( ( ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V21(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) st ASeq : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -valued bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) c= BSeq : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -valued bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) holds
(P : ( ( ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V21(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) * ASeq : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -valued bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ext-real V39() real ) set ) <= (P : ( ( ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V21(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) * BSeq : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -valued bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ext-real V39() real ) set ) ;

theorem :: PROB_3:6
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for ASeq being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for P being ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) is V71() holds
P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) is non-decreasing ;

theorem :: PROB_3:7
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for ASeq being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for P being ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) is V70() holds
P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) is non-increasing ;

definition
let X be ( ( ) ( ) set ) ;
let A1 be ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ;
func Partial_Intersection A1 -> ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ext-real ) ( ext-real ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ext-real ) ( ext-real ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ext-real ) ( ext-real ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) means :: PROB_3:def 1
( it : ( ( ) ( ) set ) . 0 : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Event of bool X : ( ( ext-real ) ( ext-real ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) = A1 : ( ( ext-real ) ( ext-real ) set ) . 0 : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Event of bool X : ( ( ext-real ) ( ext-real ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) & ( for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) holds it : ( ( ) ( ) set ) . (n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Event of bool X : ( ( ext-real ) ( ext-real ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) = (it : ( ( ) ( ) set ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) ) : ( ( ) ( ) set ) /\ (A1 : ( ( ext-real ) ( ext-real ) set ) . (n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( ) Event of bool X : ( ( ext-real ) ( ext-real ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Element of K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) );
end;

definition
let X be ( ( ) ( ) set ) ;
let A1 be ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ;
func Partial_Union A1 -> ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ext-real ) ( ext-real ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ext-real ) ( ext-real ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ext-real ) ( ext-real ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) means :: PROB_3:def 2
( it : ( ( ) ( ) set ) . 0 : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Event of bool X : ( ( ext-real ) ( ext-real ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) = A1 : ( ( ext-real ) ( ext-real ) set ) . 0 : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Event of bool X : ( ( ext-real ) ( ext-real ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) & ( for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) holds it : ( ( ) ( ) set ) . (n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Event of bool X : ( ( ext-real ) ( ext-real ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) = (it : ( ( ) ( ) set ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) ) : ( ( ) ( ) set ) \/ (A1 : ( ( ext-real ) ( ext-real ) set ) . (n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( ) Event of bool X : ( ( ext-real ) ( ext-real ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) set ) ) );
end;

theorem :: PROB_3:8
for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat)
for X being ( ( ) ( ) set )
for A1 being ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) holds (Partial_Intersection A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) c= A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) ;

theorem :: PROB_3:9
for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat)
for X being ( ( ) ( ) set )
for A1 being ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) holds A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) c= (Partial_Union A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) ;

theorem :: PROB_3:10
for X being ( ( ) ( ) set )
for A1 being ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) holds Partial_Intersection A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) is V70() ;

theorem :: PROB_3:11
for X being ( ( ) ( ) set )
for A1 being ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) holds Partial_Union A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) is V71() ;

theorem :: PROB_3:12
for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat)
for X, x being ( ( ) ( ) set )
for A1 being ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) holds
( x : ( ( ) ( ) set ) in (Partial_Intersection A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) iff for k being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) st k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) <= n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) holds
x : ( ( ) ( ) set ) in A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) ) ;

theorem :: PROB_3:13
for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat)
for X, x being ( ( ) ( ) set )
for A1 being ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) holds
( x : ( ( ) ( ) set ) in (Partial_Union A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) iff ex k being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) st
( k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) <= n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) & x : ( ( ) ( ) set ) in A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) ) ) ;

theorem :: PROB_3:14
for X being ( ( ) ( ) set )
for A1 being ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) holds Intersection (Partial_Intersection A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Element of K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) = Intersection A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Element of K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ;

theorem :: PROB_3:15
for X being ( ( ) ( ) set )
for A1 being ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) holds Union (Partial_Union A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Element of K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) = Union A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Element of K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ;

definition
let X be ( ( ) ( ) set ) ;
let A1 be ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ;
func Partial_Diff_Union A1 -> ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ext-real ) ( ext-real ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ext-real ) ( ext-real ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ext-real ) ( ext-real ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) means :: PROB_3:def 3
( it : ( ( ) ( ) set ) . 0 : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Event of bool X : ( ( ext-real ) ( ext-real ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) = A1 : ( ( ext-real ) ( ext-real ) set ) . 0 : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Event of bool X : ( ( ext-real ) ( ext-real ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) & ( for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) holds it : ( ( ) ( ) set ) . (n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Event of bool X : ( ( ext-real ) ( ext-real ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) = (A1 : ( ( ext-real ) ( ext-real ) set ) . (n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( ) Event of bool X : ( ( ext-real ) ( ext-real ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) \ ((Partial_Union A1 : ( ( ext-real ) ( ext-real ) set ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ext-real ) ( ext-real ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ext-real ) ( ext-real ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ext-real ) ( ext-real ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) ) : ( ( ) ( ) set ) : ( ( ) ( ) Element of K10(X : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) );
end;

theorem :: PROB_3:16
for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat)
for X, x being ( ( ) ( ) set )
for A1 being ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) holds
( x : ( ( ) ( ) set ) in (Partial_Diff_Union A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) iff ( x : ( ( ) ( ) set ) in A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) & ( for k being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) st k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) < n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) holds
not x : ( ( ) ( ) set ) in A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) ) ) ) ;

theorem :: PROB_3:17
for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat)
for X being ( ( ) ( ) set )
for A1 being ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) holds (Partial_Diff_Union A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) c= A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) ;

theorem :: PROB_3:18
for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat)
for X being ( ( ) ( ) set )
for A1 being ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) holds (Partial_Diff_Union A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) c= (Partial_Union A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) ;

theorem :: PROB_3:19
for X being ( ( ) ( ) set )
for A1 being ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) holds Partial_Union (Partial_Diff_Union A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) = Partial_Union A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ;

theorem :: PROB_3:20
for X being ( ( ) ( ) set )
for A1 being ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) holds Union (Partial_Diff_Union A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Element of K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) = Union A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Element of K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ;

definition
let X be ( ( ) ( ) set ) ;
let A1 be ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ;
:: original: disjoint_valued
redefine attr A1 is disjoint_valued means :: PROB_3:def 4
 for m, n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) st m : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) <> n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) holds
A1 : ( ( ext-real ) ( ext-real ) set ) . m : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) misses A1 : ( ( ext-real ) ( ext-real ) set ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) ;
end;

registration
let X be ( ( ) ( ) set ) ;
let A1 be ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ;
cluster Partial_Diff_Union A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,(bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) -> Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V73() ;
end;

registration
let X be ( ( ) ( ) set ) ;
let Si be ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) ) ;
let XSeq be ( ( Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) ) -valued bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) ) ) ;
cluster Partial_Intersection XSeq : ( ( Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,(bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) -> Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ;
end;

registration
let X be ( ( ) ( ) set ) ;
let Si be ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) ) ;
let XSeq be ( ( Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) ) -valued bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) ) ) ;
cluster Partial_Union XSeq : ( ( Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,(bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) -> Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ;
end;

registration
let X be ( ( ) ( ) set ) ;
let Si be ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) ) ;
let XSeq be ( ( Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) ) -valued bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) ) ) ;
cluster Partial_Diff_Union XSeq : ( ( Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,(bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V73() ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) -> Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ;
end;

theorem :: PROB_3:21
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) )
for YSeq, XSeq being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) st YSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = Partial_Intersection XSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) holds
( YSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = XSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) & ( for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) holds YSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = (YSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) ) : ( ( ) ( ) set ) /\ (XSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ;

theorem :: PROB_3:22
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) )
for YSeq, XSeq being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) st YSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = Partial_Union XSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) holds
( YSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = XSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) & ( for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) holds YSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = (YSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) ) : ( ( ) ( ) set ) \/ (XSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: PROB_3:23
for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat)
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) )
for XSeq being ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) ) holds (Partial_Intersection XSeq : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) c= XSeq : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) ;

theorem :: PROB_3:24
for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat)
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) )
for XSeq being ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) ) holds XSeq : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) c= (Partial_Union XSeq : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) ;

theorem :: PROB_3:25
for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat)
for X, x being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) )
for XSeq being ( ( b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) ) holds
( x : ( ( ) ( ) set ) in (Partial_Intersection XSeq : ( ( b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) iff for k being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) st k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) <= n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) holds
x : ( ( ) ( ) set ) in XSeq : ( ( b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) ) . k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) ) ;

theorem :: PROB_3:26
for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat)
for X, x being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) )
for XSeq being ( ( b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) ) holds
( x : ( ( ) ( ) set ) in (Partial_Union XSeq : ( ( b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) iff ex k being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) st
( k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) <= n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) & x : ( ( ) ( ) set ) in XSeq : ( ( b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) ) . k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) ) ) ;

theorem :: PROB_3:27
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) )
for XSeq being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) holds Partial_Intersection XSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) is V70() ;

theorem :: PROB_3:28
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) )
for XSeq being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) holds Partial_Union XSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) is V71() ;

theorem :: PROB_3:29
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) )
for XSeq being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) holds Intersection (Partial_Intersection XSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Element of K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) = Intersection XSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ;

theorem :: PROB_3:30
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) )
for XSeq being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) holds Union (Partial_Union XSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Element of K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) = Union XSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ;

theorem :: PROB_3:31
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) )
for YSeq, XSeq being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) st YSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = Partial_Diff_Union XSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V73() ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) holds
( YSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = XSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) & ( for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) holds YSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) = (XSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . (n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) \ ((Partial_Union XSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) ) : ( ( ) ( ) set ) : ( ( ) ( ) Element of K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ;

theorem :: PROB_3:32
for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat)
for X, x being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) )
for XSeq being ( ( b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) ) holds
( x : ( ( ) ( ) set ) in (Partial_Diff_Union XSeq : ( ( b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V73() ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) iff ( x : ( ( ) ( ) set ) in XSeq : ( ( b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) & ( for k being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) st k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) < n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) holds
not x : ( ( ) ( ) set ) in XSeq : ( ( b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b4 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) ) . k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) ) ) ) ;

theorem :: PROB_3:33
for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat)
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) )
for XSeq being ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) ) holds (Partial_Diff_Union XSeq : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V73() ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) c= XSeq : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) ;

theorem :: PROB_3:34
for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat)
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) )
for XSeq being ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) ) holds (Partial_Diff_Union XSeq : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V73() ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) c= (Partial_Union XSeq : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( ) ( ) set ) ) -valued bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) ;

theorem :: PROB_3:35
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) )
for XSeq being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) holds Partial_Union (Partial_Diff_Union XSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V73() ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) = Partial_Union XSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ;

theorem :: PROB_3:36
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) )
for XSeq being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) holds Union (Partial_Diff_Union XSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V73() ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Element of K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) = Union XSeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ;

theorem :: PROB_3:37
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for ASeq being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for P being ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * (Partial_Union ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) is non-decreasing ;

theorem :: PROB_3:38
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for ASeq being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for P being ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * (Partial_Intersection ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) is non-increasing ;

theorem :: PROB_3:39
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for ASeq being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for P being ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds Partial_Sums (P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) is non-decreasing ;

theorem :: PROB_3:40
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for ASeq being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for P being ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds (P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * (Partial_Union ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) . 0 : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ext-real V39() real ) Element of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) = (Partial_Sums (P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) . 0 : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ext-real V39() real ) Element of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ;

theorem :: PROB_3:41
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for ASeq being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for P being ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * (Partial_Union ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) is convergent & lim (P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * (Partial_Union ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( ext-real V39() real ) Element of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) = upper_bound (P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * (Partial_Union ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( ext-real V39() real ) Element of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) & lim (P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * (Partial_Union ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( ext-real V39() real ) Element of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) = P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (Union ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( ext-real V39() real ) set ) ) ;

theorem :: PROB_3:42
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for ASeq being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) is V73() holds
for n, m being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) st n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) < m : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) holds
(Partial_Union ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) misses ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . m : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) ;

theorem :: PROB_3:43
for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat)
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for ASeq being ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -valued bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) )
for P being ( ( ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V21(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) st ASeq : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -valued bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) is V73() holds
(P : ( ( ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V21(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) * (Partial_Union ASeq : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -valued bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -valued bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ext-real V39() real ) set ) = (Partial_Sums (P : ( ( ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) V21(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) * ASeq : ( ( b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) -valued bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b2 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ext-real V39() real ) set ) ;

theorem :: PROB_3:44
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for ASeq being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for P being ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) is V73() holds
P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * (Partial_Union ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) = Partial_Sums (P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ;

theorem :: PROB_3:45
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for ASeq being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for P being ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) is V73() holds
( Partial_Sums (P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) is convergent & lim (Partial_Sums (P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( ext-real V39() real ) Element of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) = upper_bound (Partial_Sums (P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( ext-real V39() real ) Element of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) & lim (Partial_Sums (P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( ext-real V39() real ) Element of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) = P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (Union ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( ext-real V39() real ) set ) ) ;

theorem :: PROB_3:46
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for ASeq being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for P being ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) is V73() holds
P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (Union ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( ext-real V39() real ) set ) = Sum (P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( ext-real V39() real ) Element of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ;

definition
let X be ( ( ) ( ) set ) ;
let F1 be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ;
let n be ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) ;
:: original: .
redefine func F1 . n -> ( ( ) ( ) Subset of ( ( ) ( non empty V25() ) set ) ) ;
end;

theorem :: PROB_3:47
for X being ( ( ) ( ) set ) ex F1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) st
for k being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) st k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) in dom F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( V78() V79() V80() V81() V82() V83() ) Element of K10(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) holds
F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) Subset of ( ( ) ( non empty V25() ) set ) ) = X : ( ( ) ( ) set ) ;

theorem :: PROB_3:48
for X being ( ( ) ( ) set )
for F1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) holds union (rng F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( ) Element of K10((bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( ) Element of K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) is ( ( ) ( ) Subset of ( ( ) ( non empty V25() ) set ) ) ;

definition
let X be ( ( ) ( ) set ) ;
let F1 be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ;
:: original: Union
redefine func Union F1 -> ( ( ) ( ) Subset of ( ( ) ( non empty V25() ) set ) ) ;
end;

theorem :: PROB_3:49
for X, x being ( ( ) ( ) set )
for F1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) holds
( x : ( ( ) ( ) set ) in Union F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty V25() ) set ) ) iff ex k being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) st
( k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) in dom F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( V78() V79() V80() V81() V82() V83() ) Element of K10(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) & x : ( ( ) ( ) set ) in F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) Subset of ( ( ) ( non empty V25() ) set ) ) ) ) ;

definition
let X be ( ( ) ( ) set ) ;
let F1 be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ;
func Complement F1 -> ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) means :: PROB_3:def 5
( len it : ( ( F1 : ( ( ) ( ) set ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined F1 : ( ( ) ( ) set ) -valued bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,(bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) = len F1 : ( ( ) ( ) set ) : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) & ( for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) st n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) in dom it : ( ( F1 : ( ( ) ( ) set ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined F1 : ( ( ) ( ) set ) -valued bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,(bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( V78() V79() V80() V81() V82() V83() ) Element of K10(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) holds
it : ( ( F1 : ( ( ) ( ) set ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined F1 : ( ( ) ( ) set ) -valued bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,(bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) Subset of ( ( ) ( non empty V25() ) set ) ) = (F1 : ( ( ) ( ) set ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) ) : ( ( ) ( ) Subset of ( ( ) ( non empty V25() ) set ) ) ` : ( ( ) ( ) Element of K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) );
end;

definition
let X be ( ( ) ( ) set ) ;
let F1 be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ;
func Intersection F1 -> ( ( ) ( ) Subset of ( ( ) ( non empty V25() ) set ) ) equals :: PROB_3:def 6
(Union (Complement F1 : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty V25() ) set ) ) ` : ( ( ) ( ) Element of K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) if F1 : ( ( ) ( ) set ) <> {} : ( ( ) ( ) set )
otherwise {} : ( ( ) ( ) set ) ;
end;

theorem :: PROB_3:50
for X being ( ( ) ( ) set )
for F1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) holds dom (Complement F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( V78() V79() V80() V81() V82() V83() ) Element of K10(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) = dom F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( V78() V79() V80() V81() V82() V83() ) Element of K10(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) ;

theorem :: PROB_3:51
for X, x being ( ( ) ( ) set )
for F1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) st F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) <> {} : ( ( ) ( ) set ) holds
( x : ( ( ) ( ) set ) in Intersection F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty V25() ) set ) ) iff for k being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) st k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) in dom F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( V78() V79() V80() V81() V82() V83() ) Element of K10(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) holds
x : ( ( ) ( ) set ) in F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) Subset of ( ( ) ( non empty V25() ) set ) ) ) ;

theorem :: PROB_3:52
for X, x being ( ( ) ( ) set )
for F1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) st F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) <> {} : ( ( ) ( ) set ) holds
( x : ( ( ) ( ) set ) in meet (rng F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( ) Element of K10((bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( ) Element of K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) iff for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) st n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) in dom F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( V78() V79() V80() V81() V82() V83() ) Element of K10(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) holds
x : ( ( ) ( ) set ) in F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) Subset of ( ( ) ( non empty V25() ) set ) ) ) ;

theorem :: PROB_3:53
for X being ( ( ) ( ) set )
for F1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) holds Intersection F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty V25() ) set ) ) = meet (rng F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( ) Element of K10((bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( ) Element of K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ;

theorem :: PROB_3:54
for X being ( ( ) ( ) set )
for F1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ex A1 being ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) st
( ( for k being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) st k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) in dom F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( V78() V79() V80() V81() V82() V83() ) Element of K10(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) holds
A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) = F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) Subset of ( ( ) ( non empty V25() ) set ) ) ) & ( for k being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) st not k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) in dom F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( V78() V79() V80() V81() V82() V83() ) Element of K10(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) holds
A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) = {} : ( ( ) ( ) set ) ) ) ;

theorem :: PROB_3:55
for X being ( ( ) ( ) set )
for F1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) )
for A1 being ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) st ( for k being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) st k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) in dom F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( V78() V79() V80() V81() V82() V83() ) Element of K10(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) holds
A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) = F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) Subset of ( ( ) ( non empty V25() ) set ) ) ) & ( for k being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) st not k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) in dom F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( V78() V79() V80() V81() V82() V83() ) Element of K10(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) holds
A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) = {} : ( ( ) ( ) set ) ) holds
( A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) . 0 : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Event of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) = {} : ( ( ) ( ) set ) & Union A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Element of K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) = Union F1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty V25() ) set ) ) ) ;

definition
let X be ( ( ) ( ) set ) ;
let Si be ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) ) ;
:: original: FinSequence
redefine mode FinSequence of Si -> ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Si be ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) ) ;
let FSi be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) ) ) ;
let n be ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) ;
:: original: .
redefine func FSi . n -> ( ( ) ( ) Event of Si : ( ( ) ( ) set ) ) ;
end;

theorem :: PROB_3:56
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) )
for FSi being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ex ASeq being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) st
( ( for k being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) st k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) in dom FSi : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( V78() V79() V80() V81() V82() V83() ) Element of K10(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) holds
ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) = FSi : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) ) & ( for k being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) st not k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) in dom FSi : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( V78() V79() V80() V81() V82() V83() ) Element of K10(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) holds
ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) -valued bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) . k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) = {} : ( ( ) ( ) set ) ) ) ;

theorem :: PROB_3:57
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) )
for FSi being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) holds Union FSi : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty V25() ) set ) ) in Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ;

registration
let X be ( ( ) ( ) set ) ;
let S be ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) ) ;
let F be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) ) ) ;
cluster Complement F : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) -> S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued ;
end;

theorem :: PROB_3:58
for X being ( ( ) ( ) set )
for Si being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) )
for FSi being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) holds Intersection FSi : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty V25() ) set ) ) in Si : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( ) ( ) set ) ) ;

theorem :: PROB_3:59
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for FSeq being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds dom (P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * FSeq : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( V78() V79() V80() V81() V82() V83() ) Element of K10(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) = dom FSeq : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V78() V79() V80() V81() V82() V83() ) Element of K10(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) ;

theorem :: PROB_3:60
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for FSeq being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * FSeq : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) is ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V43() V44() V45() V56() FinSequence-like FinSubsequence-like ) FinSequence of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ;

definition
let Omega be ( ( non empty ) ( non empty ) set ) ;
let Sigma be ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ;
let FSeq be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool Omega : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(Omega : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) ;
let P be ( ( ) ( non empty Relation-like Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) V21(Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) ) ) ;
:: original: *
redefine func P * FSeq -> ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V43() V44() V45() V56() FinSequence-like FinSubsequence-like ) FinSequence of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ;
end;

theorem :: PROB_3:61
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for FSeq being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds len (P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * FSeq : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V43() V44() V45() V56() FinSequence-like FinSubsequence-like ) FinSequence of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) = len FSeq : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ;

theorem :: PROB_3:62
for RFin being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V43() V44() V45() V56() FinSequence-like FinSubsequence-like ) FinSequence of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) st len RFin : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V43() V44() V45() V56() FinSequence-like FinSubsequence-like ) FinSequence of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) = 0 : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) holds
Sum RFin : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V43() V44() V45() V56() FinSequence-like FinSubsequence-like ) FinSequence of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ext-real V39() real ) Element of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) = 0 : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ;

theorem :: PROB_3:63
for RFin being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V43() V44() V45() V56() FinSequence-like FinSubsequence-like ) FinSequence of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) st len RFin : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V43() V44() V45() V56() FinSequence-like FinSubsequence-like ) FinSequence of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) >= 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) holds
ex f being ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Real_Sequence) st
( f : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Real_Sequence) . 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ext-real V39() real ) Element of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) = RFin : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V43() V44() V45() V56() FinSequence-like FinSubsequence-like ) FinSequence of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) . 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ext-real V39() real ) set ) & ( for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) st 0 : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) <> n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) & n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) < len RFin : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V43() V44() V45() V56() FinSequence-like FinSubsequence-like ) FinSequence of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) holds
f : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Real_Sequence) . (n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ext-real V39() real ) Element of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) = (f : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Real_Sequence) . n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) ) : ( ( ) ( ext-real V39() real ) set ) + (RFin : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V43() V44() V45() V56() FinSequence-like FinSubsequence-like ) FinSequence of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) . (n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) : ( ( ) ( ext-real V39() real ) set ) : ( ( ) ( ext-real V39() real ) Element of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) & Sum RFin : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V43() V44() V45() V56() FinSequence-like FinSubsequence-like ) FinSequence of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ext-real V39() real ) Element of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) = f : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Real_Sequence) . (len RFin : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V43() V44() V45() V56() FinSequence-like FinSubsequence-like ) FinSequence of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ext-real V39() real ) Element of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ;

theorem :: PROB_3:64
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for FSeq being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for ASeq being ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st ( for k being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) st k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) in dom FSeq : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V78() V79() V80() V81() V82() V83() ) Element of K10(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) holds
ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) = FSeq : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) Event of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & ( for k being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) st not k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) in dom FSeq : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V78() V79() V80() V81() V82() V83() ) Element of K10(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) holds
ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real ) Nat) : ( ( ) ( ) set ) = {} : ( ( ) ( ) set ) ) holds
( Partial_Sums (P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) is convergent & Sum (P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( ext-real V39() real ) Element of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) = (Partial_Sums (P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) . (len FSeq : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V39() real V41() V42() V78() V79() V80() V81() V82() V83() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ext-real V39() real ) Element of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) & P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (Union ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( ext-real V39() real ) set ) <= Sum (P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( ext-real V39() real ) Element of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) & Sum (P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * FSeq : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V43() V44() V45() V56() FinSequence-like FinSubsequence-like ) FinSequence of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ext-real V39() real ) Element of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) = Sum (P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * ASeq : ( ( b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -valued bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() ) Element of K10(K11(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ,REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( ext-real V39() real ) Element of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ;

theorem :: PROB_3:65
for Omega being ( ( non empty ) ( non empty ) set )
for Sigma being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of Omega : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for FSeq being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of Sigma : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (Union FSeq : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( ext-real V39() real ) set ) <= Sum (P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * FSeq : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V43() V44() V45() V56() FinSequence-like FinSubsequence-like ) FinSequence of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ext-real V39() real ) Element of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) & ( FSeq : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) is disjoint_valued implies P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (Union FSeq : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Subset of ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( ext-real V39() real ) set ) = Sum (P : ( ( ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V17(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) V21(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) V43() V44() V45() V52() ) Probability of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) * FSeq : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) -valued Function-like V43() V44() V45() V56() FinSequence-like FinSubsequence-like ) FinSequence of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ext-real V39() real ) Element of REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) ) ) ;

definition
canceled;
canceled;
let X be ( ( ) ( ) set ) ;
let IT be ( ( ) ( ) Subset-Family of ) ;
attr IT is non-decreasing-closed means :: PROB_3:def 9
 for A1 being ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) st A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) is V71() & rng A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Element of K10((bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) c= IT : ( ( ) ( ) set ) holds
Union A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Element of K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) in IT : ( ( ) ( ) set ) ;
attr IT is non-increasing-closed means :: PROB_3:def 10
 for A1 being ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) st A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) is V70() & rng A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Element of K10((bool X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) c= IT : ( ( ) ( ) set ) holds
Intersection A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Element of K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) in IT : ( ( ) ( ) set ) ;
end;

theorem :: PROB_3:66
for X being ( ( ) ( ) set )
for IT being ( ( ) ( ) Subset-Family of ) holds
( IT : ( ( ) ( ) Subset-Family of ) is non-decreasing-closed iff for A1 being ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) st A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) is V71() & rng A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Element of K10((bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) c= IT : ( ( ) ( ) Subset-Family of ) holds
lim A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Element of K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) in IT : ( ( ) ( ) Subset-Family of ) ) ;

theorem :: PROB_3:67
for X being ( ( ) ( ) set )
for IT being ( ( ) ( ) Subset-Family of ) holds
( IT : ( ( ) ( ) Subset-Family of ) is non-increasing-closed iff for A1 being ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) st A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) is V70() & rng A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Element of K10((bool b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) c= IT : ( ( ) ( ) Subset-Family of ) holds
lim A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Element of K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) in IT : ( ( ) ( ) Subset-Family of ) ) ;

theorem :: PROB_3:68
for X being ( ( ) ( ) set ) holds
( bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) is non-decreasing-closed & bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) is non-increasing-closed ) ;

registration
let X be ( ( ) ( ) set ) ;
cluster non-decreasing-closed non-increasing-closed for ( ( ) ( ) Element of K10(K10(X : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ;
end;

definition
let X be ( ( ) ( ) set ) ;
mode MonotoneClass of X is ( ( non-decreasing-closed non-increasing-closed ) ( non-decreasing-closed non-increasing-closed ) Subset-Family of ) ;
end;

theorem :: PROB_3:69
for Z, X being ( ( ) ( ) set ) holds
( Z : ( ( ) ( ) set ) is ( ( non-decreasing-closed non-increasing-closed ) ( non-decreasing-closed non-increasing-closed ) MonotoneClass of X : ( ( ) ( ) set ) ) iff ( Z : ( ( ) ( ) set ) c= bool X : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) & ( for A1 being ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) st A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) is monotone & rng A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Element of K10((bool b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( non empty V25() ) set ) ) c= Z : ( ( ) ( ) set ) holds
lim A1 : ( ( Function-like V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V17( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) V21( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) , bool b2 : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) ) SetSequence of ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) ) : ( ( ) ( ) Element of K10(b2 : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) in Z : ( ( ) ( ) set ) ) ) ) ;

theorem :: PROB_3:70
for X being ( ( ) ( ) set )
for F being ( ( non empty V25() compl-closed ) ( non empty V25() compl-closed ) Field_Subset of X : ( ( ) ( ) set ) ) holds
( F : ( ( non empty V25() compl-closed ) ( non empty V25() compl-closed ) Field_Subset of b1 : ( ( ) ( ) set ) ) is ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) SigmaField of X : ( ( ) ( ) set ) ) iff F : ( ( non empty V25() compl-closed ) ( non empty V25() compl-closed ) Field_Subset of b1 : ( ( ) ( ) set ) ) is ( ( non-decreasing-closed non-increasing-closed ) ( non-decreasing-closed non-increasing-closed ) MonotoneClass of X : ( ( ) ( ) set ) ) ) ;

theorem :: PROB_3:71
for Omega being ( ( non empty ) ( non empty ) set ) holds bool Omega : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) is ( ( non-decreasing-closed non-increasing-closed ) ( non-decreasing-closed non-increasing-closed ) MonotoneClass of Omega : ( ( non empty ) ( non empty ) set ) ) ;

definition
let Omega be ( ( non empty ) ( non empty ) set ) ;
let X be ( ( ) ( ) Subset-Family of ) ;
func monotoneclass X -> ( ( non-decreasing-closed non-increasing-closed ) ( non-decreasing-closed non-increasing-closed ) MonotoneClass of Omega : ( ( ) ( ) set ) ) means :: PROB_3:def 11
( X : ( ( ) ( ) set ) c= it : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool Omega : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of X : ( ( ) ( ) set ) ) & ( for Z being ( ( ) ( ) set ) st X : ( ( ) ( ) set ) c= Z : ( ( ) ( ) set ) & Z : ( ( ) ( ) set ) is ( ( non-decreasing-closed non-increasing-closed ) ( non-decreasing-closed non-increasing-closed ) MonotoneClass of Omega : ( ( ) ( ) set ) ) holds
it : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V78() V79() V80() V81() V82() V83() V84() ) Element of K10(REAL : ( ( ) ( non empty V56() V78() V79() V80() V84() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -defined bool Omega : ( ( ) ( ) set ) : ( ( ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(Omega : ( ( ) ( ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) -valued Function-like V56() FinSequence-like FinSubsequence-like ) FinSequence of X : ( ( ) ( ) set ) ) c= Z : ( ( ) ( ) set ) ) );
end;

theorem :: PROB_3:72
for Omega being ( ( non empty ) ( non empty ) set )
for Z being ( ( non empty V25() compl-closed ) ( non empty V25() compl-closed ) Field_Subset of Omega : ( ( non empty ) ( non empty ) set ) ) holds monotoneclass Z : ( ( non empty V25() compl-closed ) ( non empty V25() compl-closed ) Field_Subset of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non-decreasing-closed non-increasing-closed ) ( non-decreasing-closed non-increasing-closed ) MonotoneClass of b1 : ( ( non empty ) ( non empty ) set ) ) is ( ( non empty V25() compl-closed ) ( non empty V25() compl-closed ) Field_Subset of Omega : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: PROB_3:73
for Omega being ( ( non empty ) ( non empty ) set )
for Z being ( ( non empty V25() compl-closed ) ( non empty V25() compl-closed ) Field_Subset of Omega : ( ( non empty ) ( non empty ) set ) ) holds sigma Z : ( ( non empty V25() compl-closed ) ( non empty V25() compl-closed ) Field_Subset of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V25() compl-closed sigma-multiplicative ) Element of K10(K10(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V25() ) set ) ) : ( ( ) ( non empty V25() ) set ) ) = monotoneclass Z : ( ( non empty V25() compl-closed ) ( non empty V25() compl-closed ) Field_Subset of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( non-decreasing-closed non-increasing-closed ) ( non-decreasing-closed non-increasing-closed ) MonotoneClass of b1 : ( ( non empty ) ( non empty ) set ) ) ;