for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat)
for a being ( ( Function-like V18([:(Seg b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( finite b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) ,(Seg b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( finite b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) :] : ( ( ) ( Relation-like RAT : ( ( ) ( non empty non trivial non finite V81() V82() V83() V84() V87() ) set ) -valued INT : ( ( ) ( non empty non trivial non finite V81() V82() V83() V84() V85() V87() ) set ) -valued finite V61() V71() V72() V73() V74() ) set ) , REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) ) ( Relation-like [:(Seg b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( finite b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) ,(Seg b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( finite b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) :] : ( ( ) ( Relation-like RAT : ( ( ) ( non empty non trivial non finite V81() V82() V83() V84() V87() ) set ) -valued INT : ( ( ) ( non empty non trivial non finite V81() V82() V83() V84() V85() V87() ) set ) -valued finite V61() V71() V72() V73() V74() ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like total V18([:(Seg b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( finite b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) ,(Seg b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( finite b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) :] : ( ( ) ( Relation-like RAT : ( ( ) ( non empty non trivial non finite V81() V82() V83() V84() V87() ) set ) -valued INT : ( ( ) ( non empty non trivial non finite V81() V82() V83() V84() V85() V87() ) set ) -valued finite V61() V71() V72() V73() V74() ) set ) , REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) finite V61() V71() V72() V73() ) Function of [:(Seg n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( finite b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) ,(Seg m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( finite b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) :] : ( ( ) ( Relation-like RAT : ( ( ) ( non empty non trivial non finite V81() V82() V83() V84() V87() ) set ) -valued INT : ( ( ) ( non empty non trivial non finite V81() V82() V83() V84() V85() V87() ) set ) -valued finite V61() V71() V72() V73() V74() ) set ) , REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) )
for p, q being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V71() V72() V73() ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) st dom p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V71() V72() V73() ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) : ( ( ) ( finite V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) = Seg n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) : ( ( ) ( finite b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) & ( for i being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) st i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) in dom p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V71() V72() V73() ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) : ( ( ) ( finite V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) holds
ex r being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V71() V72() V73() ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) st
( dom r : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V71() V72() V73() ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) : ( ( ) ( finite V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) = Seg m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) : ( ( ) ( finite b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) & p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V71() V72() V73() ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) . i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) : ( ( ) ( V36() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) = Sum r : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V71() V72() V73() ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) : ( ( ) ( V36() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) & ( for j being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) st j : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) in dom r : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V71() V72() V73() ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) : ( ( ) ( finite V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) holds
r : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V71() V72() V73() ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) . j : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) : ( ( ) ( V36() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) = a : ( ( Function-like V18([:(Seg b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( finite b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) ,(Seg b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( finite b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) :] : ( ( ) ( Relation-like RAT : ( ( ) ( non empty non trivial non finite V81() V82() V83() V84() V87() ) set ) -valued INT : ( ( ) ( non empty non trivial non finite V81() V82() V83() V84() V85() V87() ) set ) -valued finite V61() V71() V72() V73() V74() ) set ) , REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) ) ( Relation-like [:(Seg b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( finite b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) ,(Seg b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( finite b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) :] : ( ( ) ( Relation-like RAT : ( ( ) ( non empty non trivial non finite V81() V82() V83() V84() V87() ) set ) -valued INT : ( ( ) ( non empty non trivial non finite V81() V82() V83() V84() V85() V87() ) set ) -valued finite V61() V71() V72() V73() V74() ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like total V18([:(Seg b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( finite b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) ,(Seg b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( finite b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) :] : ( ( ) ( Relation-like RAT : ( ( ) ( non empty non trivial non finite V81() V82() V83() V84() V87() ) set ) -valued INT : ( ( ) ( non empty non trivial non finite V81() V82() V83() V84() V85() V87() ) set ) -valued finite V61() V71() V72() V73() V74() ) set ) , REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) finite V61() V71() V72() V73() ) Function of [:(Seg b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( finite b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) ,(Seg b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( finite b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) :] : ( ( ) ( Relation-like RAT : ( ( ) ( non empty non trivial non finite V81() V82() V83() V84() V87() ) set ) -valued INT : ( ( ) ( non empty non trivial non finite V81() V82() V83() V84() V85() V87() ) set ) -valued finite V61() V71() V72() V73() V74() ) set ) , REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) . [i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ,j : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ] : ( ( ) ( ) set ) : ( ( ) ( V36() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) ) ) ) & dom q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V71() V72() V73() ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) : ( ( ) ( finite V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) = Seg m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) : ( ( ) ( finite b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) & ( for j being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) st j : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) in dom q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V71() V72() V73() ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) : ( ( ) ( finite V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) holds
ex s being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V71() V72() V73() ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) st
( dom s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V71() V72() V73() ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) : ( ( ) ( finite V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) = Seg n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) : ( ( ) ( finite b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) & q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V71() V72() V73() ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) . j : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) : ( ( ) ( V36() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) = Sum s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V71() V72() V73() ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) : ( ( ) ( V36() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) & ( for i being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) st i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) in dom s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V71() V72() V73() ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) : ( ( ) ( finite V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) holds
s : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V71() V72() V73() ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) . i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) : ( ( ) ( V36() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) = a : ( ( Function-like V18([:(Seg b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( finite b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) ,(Seg b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( finite b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) :] : ( ( ) ( Relation-like RAT : ( ( ) ( non empty non trivial non finite V81() V82() V83() V84() V87() ) set ) -valued INT : ( ( ) ( non empty non trivial non finite V81() V82() V83() V84() V85() V87() ) set ) -valued finite V61() V71() V72() V73() V74() ) set ) , REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) ) ( Relation-like [:(Seg b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( finite b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) ,(Seg b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( finite b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) :] : ( ( ) ( Relation-like RAT : ( ( ) ( non empty non trivial non finite V81() V82() V83() V84() V87() ) set ) -valued INT : ( ( ) ( non empty non trivial non finite V81() V82() V83() V84() V85() V87() ) set ) -valued finite V61() V71() V72() V73() V74() ) set ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like total V18([:(Seg b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( finite b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) ,(Seg b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( finite b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) :] : ( ( ) ( Relation-like RAT : ( ( ) ( non empty non trivial non finite V81() V82() V83() V84() V87() ) set ) -valued INT : ( ( ) ( non empty non trivial non finite V81() V82() V83() V84() V85() V87() ) set ) -valued finite V61() V71() V72() V73() V74() ) set ) , REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) finite V61() V71() V72() V73() ) Function of [:(Seg b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( finite b₁ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) ,(Seg b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( finite b₂ : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) -element V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) :] : ( ( ) ( Relation-like RAT : ( ( ) ( non empty non trivial non finite V81() V82() V83() V84() V87() ) set ) -valued INT : ( ( ) ( non empty non trivial non finite V81() V82() V83() V84() V85() V87() ) set ) -valued finite V61() V71() V72() V73() V74() ) set ) , REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) . [i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ,j : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ] : ( ( ) ( ) set ) : ( ( ) ( V36() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) ) ) ) holds
Sum p : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V71() V72() V73() ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) : ( ( ) ( V36() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) = Sum q : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V71() V72() V73() ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) : ( ( ) ( V36() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) ;

for F being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined ExtREAL : ( ( ) ( non empty V82() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V72() ) FinSequence of ExtREAL : ( ( ) ( non empty V82() ) set ) )
for f being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V71() V72() V73() ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) st F : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined ExtREAL : ( ( ) ( non empty V82() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V72() ) FinSequence of ExtREAL : ( ( ) ( non empty V82() ) set ) ) = f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V71() V72() V73() ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) holds
Sum F : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined ExtREAL : ( ( ) ( non empty V82() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V72() ) FinSequence of ExtREAL : ( ( ) ( non empty V82() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V82() ) set ) ) = Sum f : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V71() V72() V73() ) FinSequence of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) : ( ( ) ( V36() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) ) ;

for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V82() ) set ) -valued Function-like V72() ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V82() ) set ) -valued Function-like V72() ) PartFunc of ,) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) holds
ex F being ( ( disjoint_valued ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() disjoint_valued ) Finite_Sep_Sequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ex a being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined ExtREAL : ( ( ) ( non empty V82() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V72() ) FinSequence of ExtREAL : ( ( ) ( non empty V82() ) set ) ) st
( dom f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V82() ) set ) -valued Function-like V72() ) PartFunc of ,) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = union (rng F : ( ( disjoint_valued ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() disjoint_valued ) Finite_Sep_Sequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( finite V61() ) Element of bool b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) & dom F : ( ( disjoint_valued ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() disjoint_valued ) Finite_Sep_Sequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( finite V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) = dom a : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined ExtREAL : ( ( ) ( non empty V82() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V72() ) FinSequence of ExtREAL : ( ( ) ( non empty V82() ) set ) ) : ( ( ) ( finite V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) st n : ( ( ) ( ) set ) in dom F : ( ( disjoint_valued ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() disjoint_valued ) Finite_Sep_Sequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( finite V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) holds
for x being ( ( ) ( ) set ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined ExtREAL : ( ( ) ( non empty V82() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V72() ) FinSequence of ExtREAL : ( ( ) ( non empty V82() ) set ) ) in F : ( ( disjoint_valued ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() disjoint_valued ) Finite_Sep_Sequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( ) ( ) set ) : ( ( ) ( ) set ) holds
f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V82() ) set ) -valued Function-like V72() ) PartFunc of ,) . x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined ExtREAL : ( ( ) ( non empty V82() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V72() ) FinSequence of ExtREAL : ( ( ) ( non empty V82() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V82() ) set ) ) = a : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined ExtREAL : ( ( ) ( non empty V82() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V72() ) FinSequence of ExtREAL : ( ( ) ( non empty V82() ) set ) ) . n : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V82() ) set ) ) ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom f : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V82() ) set ) -valued Function-like V72() ) PartFunc of ,) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
ex ax being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined ExtREAL : ( ( ) ( non empty V82() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V72() ) FinSequence of ExtREAL : ( ( ) ( non empty V82() ) set ) ) st
( dom ax : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined ExtREAL : ( ( ) ( non empty V82() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V72() ) FinSequence of ExtREAL : ( ( ) ( non empty V82() ) set ) ) : ( ( ) ( finite V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) = dom a : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined ExtREAL : ( ( ) ( non empty V82() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V72() ) FinSequence of ExtREAL : ( ( ) ( non empty V82() ) set ) ) : ( ( ) ( finite V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) in dom ax : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined ExtREAL : ( ( ) ( non empty V82() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V72() ) FinSequence of ExtREAL : ( ( ) ( non empty V82() ) set ) ) : ( ( ) ( finite V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) holds
ax : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined ExtREAL : ( ( ) ( non empty V82() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V72() ) FinSequence of ExtREAL : ( ( ) ( non empty V82() ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V82() ) set ) ) = (a : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined ExtREAL : ( ( ) ( non empty V82() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() V72() ) FinSequence of ExtREAL : ( ( ) ( non empty V82() ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V82() ) set ) ) * ((chi ((F : ( ( disjoint_valued ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() disjoint_valued ) Finite_Sep_Sequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( ) set ) ,X : ( ( non empty ) ( non empty ) set ) )) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V82() ) set ) -valued Function-like V72() ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V82() ) set ) :] : ( ( ) ( Relation-like non empty V72() ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V82() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V82() ) set ) ) ) ) ) ) ;

for X being ( ( ) ( ) set )
for F being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined b₁ : ( ( ) ( ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() ) FinSequence of X : ( ( ) ( ) set ) ) holds
( F : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined b₁ : ( ( ) ( ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() ) FinSequence of b₁ : ( ( ) ( ) set ) ) is disjoint_valued iff for i, j being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) st i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) in dom F : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined b₁ : ( ( ) ( ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() ) FinSequence of b₁ : ( ( ) ( ) set ) ) : ( ( ) ( finite V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) & j : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) in dom F : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined b₁ : ( ( ) ( ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() ) FinSequence of b₁ : ( ( ) ( ) set ) ) : ( ( ) ( finite V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) & i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) <> j : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) holds
F : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined b₁ : ( ( ) ( ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() ) FinSequence of b₁ : ( ( ) ( ) set ) ) . i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) : ( ( ) ( ) set ) misses F : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined b₁ : ( ( ) ( ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() ) FinSequence of b₁ : ( ( ) ( ) set ) ) . j : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) : ( ( ) ( ) set ) ) ;

for X being ( ( non empty ) ( non empty ) set )
for A being ( ( ) ( ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for F being ( ( disjoint_valued ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined b₃ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() disjoint_valued ) Finite_Sep_Sequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) )
for G being ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined b₃ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() ) FinSequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) st dom G : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined b₃ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() ) FinSequence of b₃ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( finite V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) = dom F : ( ( disjoint_valued ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined b₃ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() disjoint_valued ) Finite_Sep_Sequence of b₃ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( finite V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) & ( for i being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) st i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) in dom G : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined b₃ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() ) FinSequence of b₃ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( finite V61() V81() V82() V83() V84() V85() V86() ) Element of bool NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) : ( ( ) ( non empty non trivial non finite ) set ) ) holds
G : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined b₃ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() ) FinSequence of b₃ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) : ( ( ) ( ) set ) = A : ( ( ) ( ) set ) /\ (F : ( ( disjoint_valued ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined b₃ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() disjoint_valued ) Finite_Sep_Sequence of b₃ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V36() real ext-real non negative finite cardinal V61() ) Nat) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) holds
G : ( ( ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined b₃ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() ) FinSequence of b₃ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) is ( ( disjoint_valued ) ( Relation-like NAT : ( ( ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V61() V62() V81() V82() V83() V84() V85() V86() V87() ) Element of bool REAL : ( ( ) ( non empty non trivial non finite V81() V82() V83() V87() ) set ) : ( ( ) ( non empty non trivial non finite ) set ) ) -defined b₃ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V61() disjoint_valued ) Finite_Sep_Sequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty V24() V25() V26() compl-closed sigma-multiplicative sigma-additive ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ;