let X be non empty set ; :: thesis: for x being Element of X
for D being set
for F being Functional_Sequence of X,COMPLEX 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
for F being Functional_Sequence of X,COMPLEX 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: for F being Functional_Sequence of X,COMPLEX 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,COMPLEX; :: 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 that
A1: F is with_the_same_dom and
A2: D c= dom (F . 0) and
A3: x in D ; :: thesis: ( Partial_Sums (F # x) is convergent iff (Partial_Sums F) # x is convergent )
A4: D c= dom ((Re F) . 0) by A2, MESFUN7C:def 11;
A5: D c= dom ((Im F) . 0) by A2, MESFUN7C:def 12;
A6: ( dom ((Partial_Sums F) . 0) = dom (F . 0) & Partial_Sums F is with_the_same_dom ) by A1, Th32, Th34;
A7: Re F is with_the_same_dom by A1;
then A8: Im F is with_the_same_dom by Th25;

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

assume A9: Partial_Sums (F # x) is convergent ; :: thesis: (Partial_Sums F) # x is convergent
then Im (Partial_Sums (F # x)) is convergent ;
then Partial_Sums (Im (F # x)) is convergent by COMSEQ_3:26;
then Partial_Sums ((Im F) # x) is convergent by A1, A2, A3, MESFUN7C:23;
then (Partial_Sums (Im F)) # x is convergent by A3, A8, A5, Th13;
then (Im (Partial_Sums F)) # x is convergent by Th29;
then A10: Im ((Partial_Sums F) # x) is convergent by A2, A3, A6, MESFUN7C:23;
Re (Partial_Sums (F # x)) is convergent by A9;
then Partial_Sums (Re (F # x)) is convergent by COMSEQ_3:26;
then Partial_Sums ((Re F) # x) is convergent by A1, A2, A3, MESFUN7C:23;
then (Partial_Sums (Re F)) # x is convergent by A3, A7, A4, Th13;
then (Re (Partial_Sums F)) # x is convergent by Th29;
then Re ((Partial_Sums F) # x) is convergent by A2, A3, A6, MESFUN7C:23;
hence (Partial_Sums F) # x is convergent by A10, COMSEQ_3:42; :: thesis: verum

end;

assume A11: (Partial_Sums F) # x is convergent ; :: thesis: Partial_Sums (F # x) is convergent
then Im ((Partial_Sums F) # x) is convergent ;
then A12: (Im (Partial_Sums F)) # x is convergent by A2, A3, A6, MESFUN7C:23;
A13: (Im F) # x = Im (F # x) by A1, A2, A3, MESFUN7C:23;
Re ((Partial_Sums F) # x) is convergent by A11;
then A14: (Re (Partial_Sums F)) # x is convergent by A2, A3, A6, MESFUN7C:23;
( Partial_Sums ((Im F) # x) is convergent iff (Partial_Sums (Im F)) # x is convergent ) by A3, A8, A5, Th13;
then A15: Im (Partial_Sums (F # x)) is convergent by A12, A13, Th29, COMSEQ_3:26;
A16: (Re F) # x = Re (F # x) by A1, A2, A3, MESFUN7C:23;
( Partial_Sums ((Re F) # x) is convergent iff (Partial_Sums (Re F)) # x is convergent ) by A3, A7, A4, Th13;
then Re (Partial_Sums (F # x)) is convergent by A14, A16, Th29, COMSEQ_3:26;
hence Partial_Sums (F # x) is convergent by A15, COMSEQ_3:42; :: thesis: verum