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 Th24:
  for f be Functional_Sequence of X,COMPLEX, n be Nat
  holds (Re f).n = Re(f.n) & (Im f).n = Im(f.n)
proof
  let f be Functional_Sequence of X,COMPLEX;
  let n be Nat;
  dom((Re f).n) = dom(f.n) by Def11;
  then
A1: dom((Re f).n) = dom(Re(f.n)) by COMSEQ_3:def 3;
  now
    let x be Element of X;
    assume
A2: x in dom((Re f).n);
    then Re(f.n).x = Re((f.n).x) by A1,COMSEQ_3:def 3;
    then
A3: Re(f.n).x = Re((f#x).n) by Def9;
    ((Re f).n).x = (Re(f#x)).n by A2,Def11;
    hence ((Re f).n).x = Re(f.n).x by A3,COMSEQ_3:def 5;
  end;
  hence (Re f).n = Re(f.n) by A1,PARTFUN1:5;
  dom((Im f).n) = dom(f.n) by Def12;
  then
A4: dom((Im f).n) = dom(Im(f.n)) by COMSEQ_3:def 4;
  now
    let x be Element of X;
    assume
A5: x in dom((Im f).n);
    then Im(f.n).x = Im((f.n).x) by A4,COMSEQ_3:def 4;
    then
A6: Im(f.n).x = Im((f#x).n) by Def9;
    ((Im f).n).x = (Im(f#x)).n by A5,Def12;
    hence ((Im f).n).x = Im(f.n).x by A6,COMSEQ_3:def 6;
  end;
  hence thesis by A4,PARTFUN1:5;
end;
