let X be non empty set ; :: thesis: for F being Functional_Sequence of X,COMPLEX holds

( F is with_the_same_dom iff Re F is with_the_same_dom )

let F be Functional_Sequence of X,COMPLEX; :: thesis: ( F is with_the_same_dom iff Re F is with_the_same_dom )

thus ( F is with_the_same_dom implies Re F is with_the_same_dom ) ; :: thesis: ( Re F is with_the_same_dom implies F is with_the_same_dom )

assume A1: Re F is with_the_same_dom ; :: thesis: F is with_the_same_dom

( F is with_the_same_dom iff Re F is with_the_same_dom )

let F be Functional_Sequence of X,COMPLEX; :: thesis: ( F is with_the_same_dom iff Re F is with_the_same_dom )

thus ( F is with_the_same_dom implies Re F is with_the_same_dom ) ; :: thesis: ( Re F is with_the_same_dom implies F is with_the_same_dom )

assume A1: Re F is with_the_same_dom ; :: thesis: F is with_the_same_dom

now :: thesis: for n, m being Nat holds dom (F . n) = dom (F . m)

hence
F is with_the_same_dom
; :: thesis: verumlet n, m be Nat; :: thesis: dom (F . n) = dom (F . m)

( dom ((Re F) . n) = dom (F . n) & dom ((Re F) . m) = dom (F . m) ) by MESFUN7C:def 11;

hence dom (F . n) = dom (F . m) by A1; :: thesis: verum

end;( dom ((Re F) . n) = dom (F . n) & dom ((Re F) . m) = dom (F . m) ) by MESFUN7C:def 11;

hence dom (F . n) = dom (F . m) by A1; :: thesis: verum