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 Th32:
  Re(f(#)g) = Re(f)(#)Re(g) - Im(f)(#)Im(g) & Im(f(#)g) = Im(f)(#)
  Re(g) + Re(f)(#)Im(g)
proof
A1: dom(Re(f)(#)Re(g)) = dom Re(f) /\ dom Re(g) by VALUED_1:def 4;
A2: dom(Im(f)(#)Im(g)) = dom Im(f) /\ dom Im(g) by VALUED_1:def 4;
A3: dom Re f = dom f by COMSEQ_3:def 3;
A4: dom Im g = dom g by COMSEQ_3:def 4;
A5: dom Re g = dom g by COMSEQ_3:def 3;
A6: dom(Re(f(#)g)) = dom(f(#)g) by COMSEQ_3:def 3;
  then
A7: dom(Re(f(#)g)) = dom f /\ dom g by VALUED_1:def 4;
A8: dom Im f = dom f by COMSEQ_3:def 4;
A9: dom(Re(f)(#)Re(g) - Im(f)(#)Im(g)) = dom(Re(f)(#)Re(g)) /\ dom(Im(f)(#)Im
  (g)) by VALUED_1:12;
  now
    let x be object;
    assume
A10: 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 A6,A10,VALUED_1:def 4;
    then
A11: Re(f(#)g).x = Re(f.x) * Re(g.x) - Im(f.x) * Im(g.x) by COMPLEX1:9;
    x in dom g by A7,A10,XBOOLE_0:def 4;
    then
A12: (Re g).x = Re(g.x) & (Im g).x = Im(g.x) by A5,A4,COMSEQ_3:def 3,def 4;
    x in dom f by A7,A10,XBOOLE_0:def 4;
    then (Re f).x = Re(f.x) & (Im f).x = Im(f.x) by A3,A8,COMSEQ_3:def 3,def 4;
    then
    Re(f(#)g).x = ((Re f)(#)(Re g)).x - (Im f).x * (Im g).x by A7,A1,A3,A5,A10
,A11,A12,VALUED_1:def 4;
    then
    Re(f(#)g).x = ((Re f)(#)(Re g)).x - ((Im f)(#)(Im g)).x by A7,A2,A8,A4,A10,
VALUED_1:def 4;
    hence
    Re(f(#)g).x = ( Re(f)(#)Re(g) - Im(f)(#)Im(g) ).x by A7,A9,A1,A2,A3,A8,A5
,A4,A10,VALUED_1:13;
  end;
  hence Re(f(#)g) = Re(f)(#)Re(g) - Im(f)(#)Im(g) by A7,A9,A1,A2,A3,A8,A5,A4,
FUNCT_1:2;
A13: dom(Im(f)(#)Re(g)) = dom Im(f) /\ dom Re(g) by VALUED_1:def 4;
A14: dom(Re(f)(#)Im(g)) = dom Re(f) /\ dom Im(g) by VALUED_1:def 4;
A15: dom(Im(f(#)g)) = dom(f(#)g) by COMSEQ_3:def 4;
  then
A16: dom(Im(f(#)g)) = dom f /\ dom g by VALUED_1:def 4;
A17: dom(Im(f)(#)Re(g) + Re(f)(#)Im(g)) = dom(Im(f)(#)Re(g)) /\ dom(Re(f)(#)Im
  (g)) by VALUED_1:def 1;
  now
    let x be object;
    assume
A18: 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 A15,A18,VALUED_1:def 4;
    then
A19: Im(f(#)g).x = Im(f.x) * Re(g.x) + Re(f.x) * Im(g.x) by COMPLEX1:9;
    x in dom g by A16,A18,XBOOLE_0:def 4;
    then
A20: Re(g).x = Re(g.x) & Im(g).x = Im(g.x) by A5,A4,COMSEQ_3:def 3,def 4;
    x in dom f by A16,A18,XBOOLE_0:def 4;
    then Re(f).x = Re(f.x) & Im(f).x = Im(f.x) by A3,A8,COMSEQ_3:def 3,def 4;
    then
    Im(f(#)g).x = ((Im f)(#)(Re g)).x + (Re f).x * (Im g).x by A16,A13,A8,A5
,A18,A19,A20,VALUED_1:def 4;
    then
    Im(f(#)g).x = ((Im f)(#)(Re g)).x + ((Re f)(#)(Im g)).x by A16,A14,A3,A4
,A18,VALUED_1:def 4;
    hence
    Im(f(#)g).x = ( Im(f)(#)Re(g) + Re(f)(#)Im(g) ).x by A16,A17,A13,A14,A3,A8
,A5,A4,A18,VALUED_1:def 1;
  end;
  hence thesis by A16,A17,A13,A14,A3,A8,A5,A4,FUNCT_1:2;
end;
