let X be non empty set ; :: thesis: for F being Functional_Sequence of X,REAL
for x being Element of X
for D being set st F is with_the_same_dom & D c= dom (F . 0 ) & x in D holds
( Partial_Sums (F # x) is convergent iff (Partial_Sums F) # x is convergent )
let F be Functional_Sequence of X,REAL ; :: thesis: for x being Element of X
for D being set st F is with_the_same_dom & D c= dom (F . 0 ) & x in D holds
( Partial_Sums (F # x) is convergent iff (Partial_Sums F) # x is convergent )
let x be Element of X; :: thesis: for D being set st F is with_the_same_dom & D c= dom (F . 0 ) & x in D holds
( Partial_Sums (F # x) is convergent iff (Partial_Sums F) # x is convergent )
let D be set ; :: thesis: ( F is with_the_same_dom & D c= dom (F . 0 ) & x in D implies ( Partial_Sums (F # x) is convergent iff (Partial_Sums F) # x is convergent ) )
assume A1: ( F is with_the_same_dom & D c= dom (F . 0 ) & x in D ) ; :: thesis: ( Partial_Sums (F # x) is convergent iff (Partial_Sums F) # x is convergent )
B2: R_EAL F is additive by Lem20;
B4: Partial_Sums (F # x) = R_EAL (Partial_Sums (F # x))
.= Partial_Sums (R_EAL (F # x)) by Lm10a ;
B5: R_EAL (F # x) = (R_EAL F) # x by MESFUN7C:1;
Partial_Sums (R_EAL F) = R_EAL (Partial_Sums F) by Lm11;
then B6: (Partial_Sums (R_EAL F)) # x = (Partial_Sums F) # x by MESFUN7C:1;

hereby :: thesis: ( (Partial_Sums F) # x is convergent implies Partial_Sums (F # x) is convergent )

assume Partial_Sums (F # x) is convergent ; :: thesis: (Partial_Sums F) # x is convergent
then Partial_Sums (R_EAL (F # x)) is convergent_to_finite_number by B4, RINFSUP2:14;
then (Partial_Sums (R_EAL F)) # x is convergent_to_finite_number by A1, B2, B5, MESFUNC9:33;
hence (Partial_Sums F) # x is convergent by B6, RINFSUP2:15; :: thesis: verum

end;

assume (Partial_Sums F) # x is convergent ; :: thesis: Partial_Sums (F # x) is convergent
then (Partial_Sums (R_EAL F)) # x is convergent_to_finite_number by B6, RINFSUP2:14;
then Partial_Sums ((R_EAL F) # x) is convergent_to_finite_number by A1, B2, MESFUNC9:33;
hence Partial_Sums (F # x) is convergent by B4, B5, RINFSUP2:15; :: thesis: verum