reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S,
  F for Functional_Sequence of X,REAL,

  f for PartFunc of X,REAL,
  seq for Real_Sequence,
  n,m for Nat,
  x for Element of X,
  z,D for set;
reserve i for Element of NAT;
reserve F for Functional_Sequence of X,COMPLEX,
  f for PartFunc of X,COMPLEX,
  A for set;

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;
