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 Th5:
  Re(f+g) = Re f + Re g & Im(f+g) = Im f + Im g
proof
A1: dom(Re(f+g)) = dom(f+g) by COMSEQ_3:def 3;
A2: dom Re g = dom g by COMSEQ_3:def 3;
A3: dom Re f = dom f by COMSEQ_3:def 3;
  then dom((Re f)+(Re g)) = dom f /\ dom g by A2,VALUED_1:def 1;
  then
A4: dom(Re(f+g)) = dom((Re f)+(Re g)) by A1,VALUED_1:def 1;
  now
    let x be object;
    assume
A5: x in dom(Re(f+g));
    then Re(f+g).x = Re((f+g).x) by COMSEQ_3:def 3;
    then Re(f+g).x = Re(f.x + g.x) by A1,A5,VALUED_1:def 1;
    then
A6: Re(f+g).x = Re(f.x) + Re(g.x) by COMPLEX1:8;
A7: x in dom f /\ dom g by A1,A5,VALUED_1:def 1;
    then x in dom Re g by A2,XBOOLE_0:def 4;
    then
A8: (Re g).x = Re(g.x) by COMSEQ_3:def 3;
    x in dom Re f by A3,A7,XBOOLE_0:def 4;
    then (Re f).x = Re(f.x) by COMSEQ_3:def 3;
    hence Re(f+g).x = (Re(f)+Re(g)).x by A4,A5,A6,A8,VALUED_1:def 1;
  end;
  hence Re(f+g) = Re(f)+Re(g) by A4,FUNCT_1:2;
A9: dom Im(f+g) = dom(f+g) by COMSEQ_3:def 4;
A10: dom Im g = dom g by COMSEQ_3:def 4;
A11: dom Im f = dom f by COMSEQ_3:def 4;
  then dom(Im f + Im g) = dom f /\ dom g by A10,VALUED_1:def 1;
  then
A12: dom Im(f+g) = dom(Im f + Im g) by A9,VALUED_1:def 1;
  now
    let x be object;
    assume
A13: x in dom Im(f+g);
    then Im(f+g).x = Im((f+g).x) by COMSEQ_3:def 4;
    then Im(f+g).x = Im(f.x + g.x) by A9,A13,VALUED_1:def 1;
    then
A14: Im(f+g).x = Im(f.x) + Im(g.x) by COMPLEX1:8;
A15: x in dom f /\ dom g by A9,A13,VALUED_1:def 1;
    then x in dom Im g by A10,XBOOLE_0:def 4;
    then
A16: (Im g).x = Im(g.x) by COMSEQ_3:def 4;
    x in dom Im f by A11,A15,XBOOLE_0:def 4;
    then (Im f).x = Im(f.x) by COMSEQ_3:def 4;
    hence Im(f+g).x = (Im(f)+Im(g)).x by A12,A13,A14,A16,VALUED_1:def 1;
  end;
  hence thesis by A12,FUNCT_1:2;
end;
