let X be non empty set ; :: thesis: for F being Functional_Sequence of X,COMPLEX holds
( Re F is with_the_same_dom iff Im F is with_the_same_dom )
let F be Functional_Sequence of X,COMPLEX ; :: thesis: ( Re F is with_the_same_dom iff Im F is with_the_same_dom )

hereby :: thesis: ( Im F is with_the_same_dom implies Re F is with_the_same_dom )

assume Re F is with_the_same_dom ; :: thesis: Im F is with_the_same_dom
then F is with_the_same_dom by Th24;
then for n, m being natural number holds dom ((Im F) . n) = dom ((Im F) . m) by MESFUNC8:def 2, MESFUN7C:def 12;
hence Im F is with_the_same_dom by MESFUNC8:def 2; :: thesis: verum

end;

assume A2: Im F is with_the_same_dom ; :: thesis: Re F is with_the_same_dom

now

let n, m be natural number ; :: thesis: dom (F . n) = dom (F . m)
( dom ((Im F) . n) = dom (F . n) & dom ((Im F) . m) = dom (F . m) ) by MESFUN7C:def 12;
hence dom (F . n) = dom (F . m) by A2, MESFUNC8:def 2; :: thesis: verum

end;

then F is with_the_same_dom by MESFUNC8:def 2;
hence Re F is with_the_same_dom ; :: thesis: verum