:: MESFUN10 semantic presentation

begin

theorem :: MESFUN10:1
for seq1, seq2 being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds seq1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) <= seq2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) holds
inf (rng seq1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) ) : ( ( non empty V50() ) ( non empty V50() V67() ) Element of K6(ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) <= inf (rng seq2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) ) : ( ( non empty V50() ) ( non empty V50() V67() ) Element of K6(ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ;

theorem :: MESFUN10:2
for seq1, seq2 being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence)
for k being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds seq1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) <= seq2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) holds
( (inferior_realsequence seq1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) ) : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued non-decreasing ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) . k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) <= (inferior_realsequence seq2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) ) : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued non-decreasing ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) . k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) & (superior_realsequence seq1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) ) : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued non-increasing ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) . k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) <= (superior_realsequence seq2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) ) : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued non-increasing ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) . k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ;

theorem :: MESFUN10:3
for seq1, seq2 being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds seq1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) <= seq2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) holds
( lim_inf seq1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) <= lim_inf seq2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) & lim_sup seq1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) <= lim_sup seq2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ;

theorem :: MESFUN10:4
for seq being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence)
for a being ( ( ) ( ext-real ) R_eal) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds seq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) >= a : ( ( ) ( ext-real ) R_eal) ) holds
inf seq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) >= a : ( ( ) ( ext-real ) R_eal) ;

theorem :: MESFUN10:5
for seq being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence)
for a being ( ( ) ( ext-real ) R_eal) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds seq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) <= a : ( ( ) ( ext-real ) R_eal) ) holds
sup seq : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) <= a : ( ( ) ( ext-real ) R_eal) ;

theorem :: MESFUN10:6
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) )
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V66() V67() V68() V69() V70() V71() V75() V76() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ) st x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in dom (inf (F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ^\ n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V66() V67() V68() V69() V70() V71() V75() V76() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) subsequence of b2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
(inf (F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ^\ n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V66() V67() V68() V69() V70() V71() V75() V76() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) subsequence of b2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) = inf ((F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ^\ n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V66() V67() V68() V69() V70() V71() V75() V76() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) subsequence of b2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # b3 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ;

theorem :: MESFUN10:7
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom (F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V29() V66() V67() V68() V69() V72() ) set ) -valued ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V66() V67() V68() V69() V70() V71() V72() V75() V76() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds
( F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) is nonnegative & F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ) holds
ex I being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) st
( ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds I : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) = Integral (M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) & Integral (M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(lim_inf F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) <= lim_inf I : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ;

begin

theorem :: MESFUN10:8
for Y being ( ( non empty ) ( non empty V67() ) Subset of ( ( ) ( ) set ) )
for r being ( ( ) ( ext-real ) R_eal) st r : ( ( ) ( ext-real ) R_eal) in REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) holds
sup ({r : ( ( ) ( ext-real ) R_eal) } : ( ( ) ( non empty V67() ) Element of K6(ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( ) set ) ) + Y : ( ( non empty ) ( non empty V67() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V67() ) Element of K6(ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) = (sup {r : ( ( ) ( ext-real ) R_eal) } : ( ( ) ( non empty V67() ) Element of K6(ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) + (sup Y : ( ( non empty ) ( non empty V67() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ;

theorem :: MESFUN10:9
for Y being ( ( non empty ) ( non empty V67() ) Subset of ( ( ) ( ) set ) )
for r being ( ( ) ( ext-real ) R_eal) st r : ( ( ) ( ext-real ) R_eal) in REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) holds
inf ({r : ( ( ) ( ext-real ) R_eal) } : ( ( ) ( non empty V67() ) Element of K6(ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( ) set ) ) + Y : ( ( non empty ) ( non empty V67() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V67() ) Element of K6(ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) = (inf {r : ( ( ) ( ext-real ) R_eal) } : ( ( ) ( non empty V67() ) Element of K6(ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) + (inf Y : ( ( non empty ) ( non empty V67() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ;

theorem :: MESFUN10:10
for seq1, seq2 being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence)
for r being ( ( ) ( ext-real ) R_eal) st r : ( ( ) ( ext-real ) R_eal) in REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds seq1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) = r : ( ( ) ( ext-real ) R_eal) + (seq2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) holds
( lim_inf seq1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) = r : ( ( ) ( ext-real ) R_eal) + (lim_inf seq2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) & lim_sup seq1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) = r : ( ( ) ( ext-real ) R_eal) + (lim_sup seq2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ;

theorem :: MESFUN10:11
for X being ( ( non empty ) ( non empty ) set )
for F1, F2 being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) st dom (F1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V29() V66() V67() V68() V69() V72() ) set ) -valued ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V66() V67() V68() V69() V70() V71() V72() V75() V76() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = dom (F2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V29() V66() V67() V68() V69() V72() ) set ) -valued ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V66() V67() V68() V69() V70() V71() V72() V75() V76() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & F1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is with_the_same_dom & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) " {+infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) } : ( ( ) ( non empty V67() ) Element of K6(ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V29() V66() V67() V68() V69() V72() ) set ) -valued ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V66() V67() V68() V69() V70() V71() V72() ) set ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) " {-infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) } : ( ( ) ( non empty V67() ) Element of K6(ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V29() V66() V67() V68() V69() V72() ) set ) -valued ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V66() V67() V68() V69() V70() V71() V72() ) set ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds F1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) = f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) + (F2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ) holds
( lim_inf F1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) = f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) + (lim_inf F2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) & lim_sup F1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) = f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) + (lim_sup F2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: MESFUN10:12
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is_integrable_on M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is_integrable_on M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) is_integrable_on M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUN10:13
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is_integrable_on M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is_integrable_on M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
ex E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st
( E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ (dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) = (Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) | E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b6 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) + (Integral (M : ( ( Function-like V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((- g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) | E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b6 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ;

theorem :: MESFUN10:14
for seq1, seq2 being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds seq1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) = - (seq2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) holds
( lim_inf seq2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) = - (lim_sup seq1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) & lim_sup seq2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) = - (lim_inf seq1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ;

theorem :: MESFUN10:15
for X being ( ( non empty ) ( non empty ) set )
for F1, F2 being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) st dom (F1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V29() V66() V67() V68() V69() V72() ) set ) -valued ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V66() V67() V68() V69() V70() V71() V72() V75() V76() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = dom (F2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V29() V66() V67() V68() V69() V72() ) set ) -valued ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V66() V67() V68() V69() V70() V71() V72() V75() V76() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & F1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is with_the_same_dom & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds F1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) = - (F2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ) holds
( lim_inf F1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) = - (lim_sup F2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) & lim_sup F1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) = - (lim_inf F2 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: MESFUN10:16
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for P being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) st E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom (F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V29() V66() V67() V68() V69() V72() ) set ) -valued ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V66() V67() V68() V69() V70() V71() V72() V75() V76() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom P : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & P : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is_integrable_on M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) )
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st x : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) in E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
|.(F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) <= P : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) . x : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) holds
( ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds |.(F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) is_integrable_on M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & |.(lim_inf F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) is_integrable_on M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & |.(lim_sup F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) is_integrable_on M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ;

theorem :: MESFUN10:17
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for P being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) st E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom (F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V29() V66() V67() V68() V69() V72() ) set ) -valued ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V66() V67() V68() V69() V70() V71() V72() V75() V76() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom P : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & P : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is_integrable_on M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & P : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is nonnegative & ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) )
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st x : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) in E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
|.(F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) <= P : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) . x : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) holds
ex I being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) st
( ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds I : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) . n : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) = Integral (M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) & lim_inf I : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) >= Integral (M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(lim_inf F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) & lim_sup I : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) <= Integral (M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(lim_sup F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) & ( ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) is convergent ) implies ( I : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) is convergent & lim I : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) = Integral (M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(lim F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ) ) ;

theorem :: MESFUN10:18
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom (F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V29() V66() V67() V68() V69() V72() ) set ) -valued ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V66() V67() V68() V69() V70() V71() V72() V75() V76() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds
( F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) is nonnegative & F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ) & ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) )
for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st x : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) in E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) <= m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds
(F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) >= (F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) & Integral (M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V29() V66() V67() V68() V69() V72() ) set ) -valued ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V66() V67() V68() V69() V70() V71() V72() V75() V76() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) | E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b5 : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) < +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) holds
ex I being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) st
( ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds I : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) = Integral (M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) & I : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) is convergent & lim I : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) = Integral (M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(lim F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ;

definition
let X be ( ( ) ( ) set ) ;
let F be ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( ) ( ) set ) ) ;
attr F is uniformly_bounded means :: MESFUN10:def 1
 ex K being ( ( real ) ( V11() real ext-real ) number ) st
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat)
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom (F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V29() V66() V67() V68() V69() V72() ) set ) -valued ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V66() V67() V68() V69() V70() V71() V72() V75() V76() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued ext-real-valued ) Element of K6(X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) holds
|.((F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) .| : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) <= K : ( ( real ) ( V11() real ext-real ) number ) ;
end;

theorem :: MESFUN10:19
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) st M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) < +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) & E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom (F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V29() V66() V67() V68() V69() V72() ) set ) -valued ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V66() V67() V68() V69() V70() V71() V72() V75() V76() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is uniformly_bounded & ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) in E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) is convergent ) holds
( ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) is_integrable_on M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & lim F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) is_integrable_on M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & ex I being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) st
( ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds I : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) = Integral (M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) & I : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) is convergent & lim I : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) = Integral (M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(lim F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ) ;

definition
let X be ( ( ) ( ) set ) ;
let F be ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( ) ( ) set ) ) ;
let f be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) ;
pred F is_uniformly_convergent_to f means :: MESFUN10:def 2
( F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) is with_the_same_dom & dom (F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V29() V66() V67() V68() V69() V72() ) set ) -valued ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V66() V67() V68() V69() V70() V71() V72() V75() V76() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued ext-real-valued ) Element of K6(X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) = dom f : ( ( Function-like V41(F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) Element of K6(K7(F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued ext-real-valued ) Element of K6(X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) & ( for e being ( ( real ) ( V11() real ext-real ) number ) st e : ( ( real ) ( V11() real ext-real ) number ) > 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V29() V66() V67() V68() V69() V72() ) set ) -valued ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V66() V67() V68() V69() V70() V71() V72() V75() V76() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ) holds
ex N being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat)
for x being ( ( ) ( ) set ) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) >= N : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) & x : ( ( ) ( ) set ) in dom (F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V29() V66() V67() V68() V69() V72() ) set ) -valued ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V66() V67() V68() V69() V70() V71() V72() V75() V76() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued ext-real-valued ) Element of K6(X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) holds
|.(((F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) - (f : ( ( Function-like V41(F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) Element of K6(K7(F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (X : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) .| : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) < e : ( ( real ) ( V11() real ext-real ) number ) ) );
end;

theorem :: MESFUN10:20
for X being ( ( non empty ) ( non empty ) set )
for F1 being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) st F1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is_uniformly_convergent_to f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) holds
for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in dom (F1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V29() V66() V67() V68() V69() V72() ) set ) -valued ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V66() V67() V68() V69() V70() V71() V72() V75() V76() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
( F1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) is convergent & lim (F1 : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) = f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ;

theorem :: MESFUN10:21
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) st M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) < +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) & E : ( ( ) ( ) Element of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom (F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V29() V66() V67() V68() V69() V72() ) set ) -valued ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V66() V67() V68() V69() V70() V71() V72() V75() V76() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) is_integrable_on M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is_uniformly_convergent_to f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is_integrable_on M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & ex I being ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) st
( ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds I : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) = Integral (M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(F : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) Element of K6(K7(b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V67() ) set ) ) : ( ( ) ( Relation-like ext-real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) & I : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) is convergent & lim I : ( ( Function-like V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V66() V67() V68() V69() V70() V71() V72() ) Element of K6(REAL : ( ( ) ( non empty V29() V66() V67() V68() V72() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) = Integral (M : ( ( Function-like V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like total V41(b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V67() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b3 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V67() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V67() ) set ) ) ) ) ;