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 )
A2: R_EAL (F # x) = (R_EAL F) # x by MESFUN7C:1;
Partial_Sums (R_EAL F) = R_EAL (Partial_Sums F) by Th7;
then A3: (Partial_Sums (R_EAL F)) # x = (Partial_Sums F) # x by MESFUN7C:1;
A4: Partial_Sums (F # x) = R_EAL (Partial_Sums (F # x))
.= Partial_Sums (R_EAL (F # x)) by Th5 ;
A5: R_EAL F is additive by Th9;

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 A4, RINFSUP2:14;
then (Partial_Sums (R_EAL F)) # x is convergent_to_finite_number by A1, A5, A2, MESFUNC9:33;
hence (Partial_Sums F) # x is convergent by A3, 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 A3, RINFSUP2:14;
then Partial_Sums ((R_EAL F) # x) is convergent_to_finite_number by A1, A5, MESFUNC9:33;
hence Partial_Sums (F # x) is convergent by A4, A2, RINFSUP2:15; :: thesis: verum