:: MESFUNC9 semantic presentation

begin

theorem :: MESFUNC9:1
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) 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 V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is V120() & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is V120() holds
dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC9:2
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) 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 V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is V120() & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is V119() holds
dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC9:3
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) 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 V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is V119() & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is V119() holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is V119() ;

theorem :: MESFUNC9:4
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) 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 V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is V120() & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is V120() holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) + g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is V120() ;

theorem :: MESFUNC9:5
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) 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 V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is V119() & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is V120() holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is V119() ;

theorem :: MESFUNC9:6
for X being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) 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 V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is V120() & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is V119() holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) - g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is V120() ;

theorem :: MESFUNC9:7
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) holds
( ( seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is convergent_to_finite_number implies ex g being ( ( real ) ( V11() real ext-real ) number ) st
( lim seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = g : ( ( real ) ( V11() real ext-real ) number ) & ( for p being ( ( real ) ( V11() real ext-real ) number ) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) < p : ( ( real ) ( V11() real ext-real ) number ) holds
ex n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st
for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st 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
|.((seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) - (lim seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) .| : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) < p : ( ( real ) ( V11() real ext-real ) number ) ) ) ) & ( seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is convergent_to_+infty implies lim seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) & ( seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is convergent_to_-infty implies lim seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = -infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) ) ;

theorem :: MESFUNC9:8
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) st seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is V111() holds
not seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is convergent_to_-infty ;

theorem :: MESFUNC9:9
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence)
for p being ( ( ext-real ) ( ext-real ) number ) st seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is convergent & ( for k being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) . k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) <= p : ( ( ext-real ) ( ext-real ) number ) ) holds
lim seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) <= p : ( ( ext-real ) ( ext-real ) number ) ;

theorem :: MESFUNC9:10
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence)
for p being ( ( ext-real ) ( ext-real ) number ) st seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is convergent & ( for k being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds p : ( ( ext-real ) ( ext-real ) number ) <= seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) . k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) holds
p : ( ( ext-real ) ( ext-real ) number ) <= lim seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;

theorem :: MESFUNC9:11
for seq1, seq2, seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) st seq1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is convergent & seq2 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is convergent & seq1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is V111() & seq2 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is V111() & ( for k being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) . k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = (seq1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) . k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) + (seq2 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) . k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) holds
( seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is V111() & seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is convergent & lim seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = (lim seq1 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) + (lim seq2 : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) ;

theorem :: MESFUNC9:12
for X being ( ( non empty ) ( non empty ) set )
for G, F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 D being ( ( ) ( ) set ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds G : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) = (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) | D : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b5 : ( ( ) ( ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) & x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in D : ( ( ) ( ) set ) holds
( ( F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is convergent_to_+infty implies G : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is convergent_to_+infty ) & ( F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is convergent_to_-infty implies G : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is convergent_to_-infty ) & ( F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is convergent_to_finite_number implies G : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is convergent_to_finite_number ) & ( F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is convergent implies G : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is convergent ) ) ;

theorem :: MESFUNC9:13
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) 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 V96() V97() V98() sigma-additive ) 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 V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) st E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is nonnegative & M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . (E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) /\ (eq_dom (f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) ,+infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) )) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) <> 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
Integral (M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;

theorem :: MESFUNC9:14
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) 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 V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( Integral (M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(chi (E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,X : ( ( non empty ) ( non empty ) set ) )) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) & Integral (M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((chi (E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,X : ( ( non empty ) ( non empty ) set ) )) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) | E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b4 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) . E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) ;

theorem :: MESFUNC9:15
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) 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 V96() V97() V98() sigma-additive ) 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 V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) st E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is nonnegative & ( 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 b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) 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 V74() ) set ) ) <= g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) 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 V74() ) set ) ) ) holds
Integral (M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) 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 V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b4 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) <= Integral (M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(g : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) 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 V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b4 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;

begin

definition
let f be ( ( Relation-like Function-like ext-real-valued ) ( Relation-like Function-like ext-real-valued ) Function) ;
let x be ( ( ) ( ) set ) ;
:: original: .
redefine func f . x -> ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;
end;

definition
let s be ( ( Relation-like Function-like ext-real-valued ) ( Relation-like Function-like ext-real-valued ) Function) ;
func Partial_Sums s -> ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) means :: MESFUNC9:def 1
( it : ( ( ) ( ) set ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = s : ( ( ) ( ) set ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds it : ( ( ) ( ) set ) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = (it : ( ( ) ( ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) + (s : ( ( ) ( ) set ) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) );
end;

definition
let s be ( ( Relation-like Function-like ext-real-valued ) ( Relation-like Function-like ext-real-valued ) Function) ;
attr s is summable means :: MESFUNC9:def 2
 Partial_Sums s : ( ( ) ( ) set ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is convergent ;
end;

definition
let s be ( ( Relation-like Function-like ext-real-valued ) ( Relation-like Function-like ext-real-valued ) Function) ;
func Sum s -> ( ( ) ( ext-real ) R_eal) equals :: MESFUNC9:def 3
 lim (Partial_Sums s : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;
end;

theorem :: MESFUNC9:16
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) st seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is V111() holds
( Partial_Sums seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is V111() & Partial_Sums seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is non-decreasing ) ;

theorem :: MESFUNC9:17
for seq being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) < seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total 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 V74() ) set ) ) ) holds
for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) < (Partial_Sums seq : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;

theorem :: MESFUNC9:18
for X being ( ( non empty ) ( non empty ) set )
for F, G being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for D being ( ( ) ( ) set ) st F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 G : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) = (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) | D : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b4 : ( ( ) ( ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) holds
G : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is with_the_same_dom ;

theorem :: MESFUNC9:19
for X being ( ( non empty ) ( non empty ) set )
for F, G being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for D being ( ( ) ( ) set ) st D : ( ( ) ( ) set ) c= dom (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds G : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) = (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) | D : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b4 : ( ( ) ( ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) & ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in D : ( ( ) ( ) set ) holds
F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is convergent ) holds
(lim F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) | D : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b4 : ( ( ) ( ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) = lim G : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC9:20
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for F, G being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is with_the_same_dom & E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & ( for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds
( F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & G : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) = (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) | E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) ) holds
G : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC9:21
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for F, G being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) st E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & G : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is with_the_same_dom & ( 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 b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is summable ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds G : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) = (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) | E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) holds
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 b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
G : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is summable ;

begin

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let F be ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) ;
func Partial_Sums F -> ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( ) ( ) set ) ) means :: MESFUNC9:def 4
( it : ( ( ) ( ) set ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) = F : ( ( ) ( ) set ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds it : ( ( ) ( ) set ) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) = (it : ( ( ) ( ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) + (F : ( ( ) ( ) set ) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) );
end;

definition
let X be ( ( ) ( ) set ) ;
let F be ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( ) ( ) set ) ) ;
attr F is additive means :: MESFUNC9:def 5
 for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st 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
for x being ( ( ) ( ) set ) holds
( not x : ( ( ) ( ) set ) in (dom (F : ( ( ) ( ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom (F : ( ( ) ( ) set ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) or (F : ( ( ) ( ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) <> +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) or (F : ( ( ) ( ) set ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) <> -infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) );
end;

theorem :: MESFUNC9:22
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat)
for z being ( ( ) ( ) set ) st z : ( ( ) ( ) set ) in dom ((Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & m : ( ( 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) holds
( z : ( ( ) ( ) set ) in dom ((Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & z : ( ( ) ( ) set ) in dom (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC9:23
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat)
for z being ( ( ) ( ) set ) st z : ( ( ) ( ) set ) in dom ((Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & ((Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . z : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) holds
ex m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st
( m : ( ( 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) & (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . z : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) ;

theorem :: MESFUNC9:24
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat)
for z being ( ( ) ( ) set ) st F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is additive & z : ( ( ) ( ) set ) in dom ((Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & ((Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . z : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) & m : ( ( 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) holds
(F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . z : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) <> -infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;

theorem :: MESFUNC9:25
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat)
for z being ( ( ) ( ) set ) st z : ( ( ) ( ) set ) in dom ((Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & ((Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . z : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = -infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) holds
ex m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st
( m : ( ( 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) & (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . z : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = -infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) ;

theorem :: MESFUNC9:26
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat)
for z being ( ( ) ( ) set ) st F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is additive & z : ( ( ) ( ) set ) in dom ((Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & ((Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . z : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = -infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) & m : ( ( 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) holds
(F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . z : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) <> +infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;

theorem :: MESFUNC9:27
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is additive holds
( (((Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) " {-infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) } : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ ((F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) " {+infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) } : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() ) set ) & (((Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) " {+infty : ( ( ) ( non empty non real ext-real positive non negative ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) } : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ ((F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V47() V73() V74() V75() V76() V77() V78() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) " {-infty : ( ( ) ( non empty non real ext-real non positive negative ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) } : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() ) set ) ) ;

theorem :: MESFUNC9:28
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is additive holds
dom ((Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = meet { (dom (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) where k is ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) : k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V47() V73() V74() V75() V76() V77() V78() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) <= n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) } : ( ( ) ( ) set ) ;

theorem :: MESFUNC9:29
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is additive & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is with_the_same_dom holds
dom ((Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = dom (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC9:30
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ) holds
F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is additive ;

theorem :: MESFUNC9:31
for X being ( ( non empty ) ( non empty ) set )
for F, G being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for D being ( ( ) ( ) set ) st F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is additive & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds G : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) = (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) | D : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b4 : ( ( ) ( ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) holds
G : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is additive ;

theorem :: MESFUNC9:32
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat)
for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) )
for D being ( ( ) ( ) set ) st F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is additive & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is with_the_same_dom & D : ( ( ) ( ) set ) c= dom (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in D : ( ( ) ( ) set ) holds
(Partial_Sums (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total 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 V74() ) set ) ) = ((Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;

theorem :: MESFUNC9:33
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 D being ( ( ) ( ) set ) st F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is additive & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is with_the_same_dom & D : ( ( ) ( ) set ) c= dom (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in D : ( ( ) ( ) set ) holds
( ( Partial_Sums (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is convergent_to_finite_number implies (Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is convergent_to_finite_number ) & ( (Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is convergent_to_finite_number implies Partial_Sums (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is convergent_to_finite_number ) & ( Partial_Sums (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is convergent_to_+infty implies (Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is convergent_to_+infty ) & ( (Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is convergent_to_+infty implies Partial_Sums (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is convergent_to_+infty ) & ( Partial_Sums (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is convergent_to_-infty implies (Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is convergent_to_-infty ) & ( (Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is convergent_to_-infty implies Partial_Sums (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is convergent_to_-infty ) & ( Partial_Sums (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is convergent implies (Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is convergent ) & ( (Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is convergent implies Partial_Sums (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is convergent ) ) ;

theorem :: MESFUNC9:34
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,)
for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is additive & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is with_the_same_dom & dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) c= dom (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is summable & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) 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 V74() ) set ) ) = Sum (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) R_eal) holds
f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) 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 V74() ) set ) ) = lim ((Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;

theorem :: MESFUNC9:35
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st ( for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is additive & (Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC9:36
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st ( for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ) holds
(Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ;

theorem :: MESFUNC9:37
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat)
for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is with_the_same_dom & x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in dom (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & ( for k being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . k : ( ( 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ) & 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
((Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) <= ((Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;

theorem :: MESFUNC9:38
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is with_the_same_dom & x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in dom (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & ( for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ) holds
( (Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is non-decreasing & (Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is convergent ) ;

theorem :: MESFUNC9:39
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st ( for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is V119() ) holds
(Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is V119() ;

theorem :: MESFUNC9:40
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st ( for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is V120() ) holds
(Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is V120() ;

theorem :: MESFUNC9:41
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for m 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
( F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is V119() ) ) holds
(Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC9:42
for X being ( ( non empty ) ( non empty ) set )
for F, G being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat)
for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is additive & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is with_the_same_dom & G : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is additive & G : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is with_the_same_dom & x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in (dom (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom (G : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & ( for k being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat)
for y being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st y : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in (dom (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (dom (G : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
(F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . k : ( ( 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . y : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) <= (G : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . k : ( ( 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . y : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) holds
((Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) <= ((Partial_Sums G : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;

theorem :: MESFUNC9:43
for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) st F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is additive & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is with_the_same_dom holds
Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is with_the_same_dom ;

theorem :: MESFUNC9:44
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) st dom (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is additive & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 (Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) 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 b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is summable ) holds
lim (Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC9:45
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) holds
for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds (Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_integrable_on M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC9:46
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) 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 V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for I being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence)
for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is additive & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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
( F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative & I : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total 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 V74() ) set ) ) = Integral (M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) ) holds
Integral (M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = (Partial_Sums I : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ;

begin

theorem :: MESFUNC9:47
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) 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 V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) st E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is nonnegative & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is additive & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds
( F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative & E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) & ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) in E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is summable & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) . x : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = Sum (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) R_eal) ) ) holds
ex I being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total 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 quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total 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 V74() ) set ) ) = Integral (M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) | E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b4 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) & I : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is summable & Integral (M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) 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 V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b4 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = Sum I : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) R_eal) ) ;

theorem :: MESFUNC9:48
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) 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 V96() V97() V98() sigma-additive ) 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 V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) st E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is nonnegative & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
ex g being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) st
( g : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is additive & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds
( g : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & g : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative & g : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ) & ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) in E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
( g : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is summable & f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) . x : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = Sum (g : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) R_eal) ) ) & ex I being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total 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 quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total 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 V74() ) set ) ) = Integral (M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((g : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) | E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b4 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) & I : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is summable & Integral (M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(f : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) 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 V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b4 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = Sum I : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) R_eal) ) ) ;

registration
let X be ( ( non empty ) ( non empty ) set ) ;
cluster non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total with_the_same_dom additive for ( ( ) ( ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let C, D, X be ( ( non empty ) ( non empty ) set ) ;
let F be ( ( Function-like quasi_total ) ( non empty Relation-like [:C : ( ( non empty ) ( non empty ) set ) ,D : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Function of [:C : ( ( non empty ) ( non empty ) set ) ,D : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ;
let c be ( ( ) ( ) Element of C : ( ( non empty ) ( non empty ) set ) ) ;
let d be ( ( ) ( ) Element of D : ( ( non empty ) ( non empty ) set ) ) ;
:: original: .
redefine func F . (c,d) -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) ;
end;

definition
let C, D, X be ( ( non empty ) ( non empty ) set ) ;
let F be ( ( Function-like quasi_total ) ( non empty Relation-like [:C : ( ( non empty ) ( non empty ) set ) ,D : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) -defined X : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of [:C : ( ( non empty ) ( non empty ) set ) ,D : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) ;
let c be ( ( ) ( ) Element of C : ( ( non empty ) ( non empty ) set ) ) ;
func ProjMap1 (F,c) -> ( ( Function-like quasi_total ) ( non empty Relation-like D : ( ( ) ( ) set ) -defined X : ( ( ) ( ) set ) -valued Function-like total quasi_total ) Function of D : ( ( ) ( ) set ) ,X : ( ( ) ( ) set ) ) means :: MESFUNC9:def 6
 for d being ( ( ) ( ) Element of D : ( ( ) ( ) set ) ) holds it : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined F : ( ( ) ( ) Element of C : ( ( non empty ) ( non empty ) set ) ) -valued Function-like ) Element of bool [:X : ( ( ) ( ) set ) ,F : ( ( ) ( ) Element of C : ( ( non empty ) ( non empty ) set ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) . d : ( ( ) ( ) Element of D : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) = F : ( ( ) ( ) Element of C : ( ( non empty ) ( non empty ) set ) ) . (c : ( ( ) ( ) Element of C : ( ( non empty ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of D : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) ;
end;

definition
let C, D, X be ( ( non empty ) ( non empty ) set ) ;
let F be ( ( Function-like quasi_total ) ( non empty Relation-like [:C : ( ( non empty ) ( non empty ) set ) ,D : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) -defined X : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of [:C : ( ( non empty ) ( non empty ) set ) ,D : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) ;
let d be ( ( ) ( ) Element of D : ( ( non empty ) ( non empty ) set ) ) ;
func ProjMap2 (F,d) -> ( ( Function-like quasi_total ) ( non empty Relation-like C : ( ( non empty ) ( non empty ) set ) -defined X : ( ( ) ( ) set ) -valued Function-like total quasi_total ) Function of C : ( ( non empty ) ( non empty ) set ) ,X : ( ( ) ( ) set ) ) means :: MESFUNC9:def 7
 for c being ( ( ) ( ) Element of C : ( ( non empty ) ( non empty ) set ) ) holds it : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined F : ( ( ) ( ) Element of C : ( ( non empty ) ( non empty ) set ) ) -valued Function-like ) Element of bool [:X : ( ( ) ( ) set ) ,F : ( ( ) ( ) Element of C : ( ( non empty ) ( non empty ) set ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) . c : ( ( ) ( ) Element of C : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) = F : ( ( ) ( ) Element of C : ( ( non empty ) ( non empty ) set ) ) . (c : ( ( ) ( ) Element of C : ( ( non empty ) ( non empty ) set ) ) ,d : ( ( ) ( ) Element of C : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) ;
end;

definition
let X, Y be ( ( ) ( ) set ) ;
let F be ( ( Function-like quasi_total ) ( non empty Relation-like [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty Relation-like RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued INT : ( ( ) ( non empty V29() V73() V74() V75() V76() V77() V79() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) -defined PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Function of [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty Relation-like RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued INT : ( ( ) ( non empty V29() V73() V74() V75() V76() V77() V79() ) set ) -valued complex-valued ext-real-valued real-valued natural-valued ) set ) , PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) ;
let n be ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ;
func ProjMap1 (F,n) -> ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) means :: MESFUNC9:def 8
 for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds it : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined Y : ( ( ) ( ) set ) -valued Function-like ) Element of bool [:X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = F : ( ( ) ( ) set ) . (n : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) ,m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( ) ( ) set ) ;
func ProjMap2 (F,n) -> ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) means :: MESFUNC9:def 9
 for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds it : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( Function-like ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined Y : ( ( ) ( ) set ) -valued Function-like ) Element of bool [:X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = F : ( ( ) ( ) set ) . (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ,n : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) ;
end;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let F be ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) ) -valued Function-like total quasi_total ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ;
let n be ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ;
:: original: .
redefine func F . n -> ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) ;
end;

theorem :: MESFUNC9:49
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) st E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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
( F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) ) -valued Function-like total quasi_total ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) ) -valued Function-like total quasi_total ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ) holds
ex FF being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) ) -valued Function-like total quasi_total ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) st
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds
( ( for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds
( (FF : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) ) -valued Function-like total quasi_total ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . m : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_simple_func_in S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) & dom ((FF : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) ) -valued Function-like total quasi_total ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . m : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = dom (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) & ( for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds (FF : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) ) -valued Function-like total quasi_total ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . m : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ) & ( for j, k being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st j : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <= k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds
for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in dom (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
((FF : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) ) -valued Function-like total quasi_total ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . j : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) <= ((FF : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) ) -valued Function-like total quasi_total ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . k : ( ( 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) & ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in dom (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
( (FF : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) ) -valued Function-like total quasi_total ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is convergent & lim ((FF : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) ) -valued Function-like total quasi_total ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , Funcs (NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) )) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) ) ) ;

theorem :: MESFUNC9:50
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) 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 V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) st E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is additive & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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
( F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative ) ) holds
ex I being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total 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 quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total 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 V74() ) set ) ) = Integral (M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) & Integral (M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = (Partial_Sums I : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total 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 V74() ) set ) ) ) ;

theorem :: MESFUNC9:51
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) 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 V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) st E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) c= dom (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) is additive & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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
( F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) ) & ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) in E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is summable ) holds
ex I being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total 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 quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total 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 V74() ) set ) ) = Integral (M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) | E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b4 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) & I : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is summable & Integral (M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,((lim (Partial_Sums F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) | E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b4 : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = Sum I : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) R_eal) ) ;

theorem :: MESFUNC9:52
for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) 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 V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) )
for F being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) st E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) = dom (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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-zero empty-yielding RAT : ( ( ) ( non empty V29() V73() V74() V75() V76() V79() ) set ) -valued Function-like one-to-one constant functional ext-real non positive non negative V47() complex-valued ext-real-valued real-valued natural-valued V73() V74() V75() V76() V77() V78() V79() V82() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is nonnegative & F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ) & ( for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st n : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) <= m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds
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 b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
(F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . n : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) <= (F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) & ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) in E : ( ( ) ( ) Element of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) holds
F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) # x : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is convergent ) holds
ex I being ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total 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 quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total 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 V74() ) set ) ) = Integral (M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) & I : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) is convergent & Integral (M : ( ( Function-like quasi_total zeroed nonnegative sigma-additive ) ( non empty Relation-like b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b2 : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative V96() V97() V98() sigma-additive ) SigmaField of b1 : ( ( non empty ) ( non empty ) set ) ) ) ,(lim F : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total ) 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 V74() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V74() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) = lim I : ( ( Function-like quasi_total ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V73() V74() V75() V76() V77() V78() V79() ) Element of bool REAL : ( ( ) ( non empty V29() V73() V74() V75() V79() ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V74() ) set ) -valued Function-like total quasi_total ext-real-valued ) ExtREAL_sequence) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V74() ) set ) ) ) ;