reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,COMPLEX,
  r for Real,
  k for Real,
  n for Nat,
  E for Element of S;

theorem Th23:
  for f be with_the_same_dom Functional_Sequence of X,COMPLEX, x
be Element of X st x in dom (f.0) holds (Re f)#x = Re(f#x) & (Im f)#x = Im(f#x)
proof
  let f be with_the_same_dom Functional_Sequence of X,COMPLEX;
  let x be Element of X;
  set F = Re f;
  set G = Im f;
  assume
A1: x in dom (f.0);
  now
    let n be Element of NAT;
    dom(F.n) = dom(f.n) by Def11;
    then
A2: dom(F.n) = dom (f.0) by MESFUNC8:def 2;
    dom(G.n) = dom(f.n) by Def12;
    then
A3: dom(G.n) = dom (f.0) by MESFUNC8:def 2;
    (F#x).n = (F.n).x & (G#x).n = (G.n).x by SEQFUNC:def 10;
    hence (F#x).n = (Re(f#x)).n & (G#x).n = (Im(f#x)).n by A1,A2,A3,Def11,Def12
;
  end;
  hence thesis by FUNCT_2:63;
end;
