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 Th3:
  Re(c(#)f) = (Re c)(#)(Re f) - (Im c)(#)(Im f) & Im(c(#)f) = (Im c
  )(#)(Re f) + (Re c)(#)(Im f)
proof
A1: dom((Re c)(#)(Re f)) = dom Re f by VALUED_1:def 5;
A2: dom((Im c)(#)(Im f)) = dom Im f by VALUED_1:def 5;
A3: dom(c(#)f) = dom f by VALUED_1:def 5;
A4: dom(Re(c(#)f)) = dom(c(#)f) by COMSEQ_3:def 3;
A5: dom Re f = dom f by COMSEQ_3:def 3;
A6: dom Im f = dom f by COMSEQ_3:def 4;
A7: dom( Re(c)(#)Re(f) - Im(c)(#)Im(f) ) = dom(Re(c)(#)Re(f)) /\ dom(Im(c)
  (#)Im(f)) by VALUED_1:12;
  now
    let x be object;
A8: Re(c * f.x) = Re(c) * Re(f.x) - Im(c) * Im(f.x) by COMPLEX1:9;
    assume
A9: x in dom(Re(c(#)f));
    then
A10: (Im(c)(#)Im(f)).x = Im(c) * Im(f).x by A4,A6,A2,A3,VALUED_1:def 5;
A11: Im(f).x = Im(f.x) by A4,A6,A3,A9,COMSEQ_3:def 4;
A12: Re(f).x = Re(f.x) by A4,A5,A3,A9,COMSEQ_3:def 3;
    Re(c(#)f).x = Re((c(#)f).x) by A9,COMSEQ_3:def 3;
    then
A13: Re(c(#)f).x = Re(c * f.x) by A4,A9,VALUED_1:def 5;
    (Re(c)(#)Re(f)).x = Re(c) * Re(f).x by A4,A5,A1,A3,A9,VALUED_1:def 5;
    hence Re(c(#)f).x = ( Re(c)(#)Re(f) - Im(c)(#)Im(f) ).x by A4,A5,A6,A1,A2
,A3,A7,A9,A13,A10,A12,A11,A8,VALUED_1:13;
  end;
  hence Re(c(#)f) = Re(c)(#)Re(f) - Im(c)(#)Im(f) by A4,A5,A6,A1,A2,A3,A7,
FUNCT_1:2;
A14: dom((Im c)(#)(Re f)) = dom Re f by VALUED_1:def 5;
A15: dom((Re c)(#)(Im f)) = dom Im f by VALUED_1:def 5;
A16: dom(Im(c(#)f)) = dom(c(#)f) by COMSEQ_3:def 4;
A17: dom( Im(c)(#)Re(f) + Re(c)(#)Im(f) ) = dom(Im(c)(#)Re(f)) /\ dom(Re(c)
  (#)Im(f)) by VALUED_1:def 1;
  now
    let x be object;
    assume
A18: x in dom(Im(c(#)f));
    then
A19: (Im(c)(#)Re(f)).x = Im(c) * Re(f).x by A5,A16,A14,A3,VALUED_1:def 5;
    Im(c(#)f).x = Im((c(#)f).x) by A18,COMSEQ_3:def 4;
    then
A20: Im(c(#)f).x = Im(c * f.x) by A16,A18,VALUED_1:def 5;
A21: (Im f).x = Im(f.x) by A16,A6,A3,A18,COMSEQ_3:def 4;
A22: (Re f).x = Re(f.x) by A5,A16,A3,A18,COMSEQ_3:def 3;
A23: (Re(c)(#)Im(f)).x = Re(c) * Im(f).x by A16,A6,A15,A3,A18,VALUED_1:def 5;
    (Im(c)(#)Re(f) + Re(c)(#)Im(f)).x = (Im(c)(#)Re(f)).x + (Re(c)(#)Im(f
    )). x by A5,A16,A6,A14,A15,A3,A17,A18,VALUED_1:def 1;
    hence Im(c(#)f).x = ( Im(c)(#)Re(f) + Re(c)(#)Im(f) ).x by A20,A19,A23,A22
,A21,COMPLEX1:9;
  end;
  hence thesis by A5,A16,A6,A14,A15,A3,A17,FUNCT_1:2;
end;
