theorem Th24:
  F is with_the_same_dom iff Re F is with_the_same_dom
proof
  thus F is with_the_same_dom implies Re F is with_the_same_dom;
  assume
A1: Re F is with_the_same_dom;
  now
    let n,m be Nat;
    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;
  end;
  hence F is with_the_same_dom;
end;
