let X be non empty set ; :: thesis: for n being Nat
for F being Functional_Sequence of X,COMPLEX st F is with_the_same_dom holds
dom ((Partial_Sums F) . n) = dom (F . 0)
let n be Nat; :: thesis: for F being Functional_Sequence of X,COMPLEX st F is with_the_same_dom holds
dom ((Partial_Sums F) . n) = dom (F . 0)
let F be Functional_Sequence of X,COMPLEX; :: thesis: ( F is with_the_same_dom implies dom ((Partial_Sums F) . n) = dom (F . 0) )
assume F is with_the_same_dom ; :: thesis: dom ((Partial_Sums F) . n) = dom (F . 0)
then Re F is with_the_same_dom ;
then dom ((Partial_Sums (Re F)) . n) = dom ((Re F) . 0) by Th11;
then dom ((Partial_Sums (Re F)) . n) = dom (F . 0) by MESFUN7C:def 11;
then dom ((Re (Partial_Sums F)) . n) = dom (F . 0) by Th29;
hence dom ((Partial_Sums F) . n) = dom (F . 0) by MESFUN7C:def 11; :: thesis: verum