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,
  c for Complex,
  E,A,B for Element of S;

theorem Th8:
  f = Re f + <i>(#)(Im f)
proof
A1: dom f = dom Re f by COMSEQ_3:def 3;
A2: dom f = dom Im f by COMSEQ_3:def 4;
A3: dom(Re f + <i>(#)(Im f)) = dom Re f /\ dom(<i>(#)(Im f)) by VALUED_1:def 1
    .= dom f /\ dom f by A1,A2,VALUED_1:def 5;
A4: dom(<i>(#)(Im f)) = dom Im f by VALUED_1:def 5;
  now
    let x be object;
    assume
A5: x in dom(Re f + <i>(#)(Im f));
    then (Re f + <i>(#)(Im f)).x = (Re f).x + (<i>(#)(Im f)).x by
VALUED_1:def 1
      .= Re(f.x) + (<i>(#)(Im f)).x by A1,A3,A5,COMSEQ_3:def 3
      .= Re(f.x) + <i> * (Im f).x by A2,A4,A3,A5,VALUED_1:def 5
      .= Re(f.x) + <i> * Im(f.x) by A2,A3,A5,COMSEQ_3:def 4;
    hence (Re f + <i>(#)(Im f)).x = f.x by COMPLEX1:13;
  end;
  hence thesis by A3,FUNCT_1:2;
end;
