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 Th2:
  r(#)(Re f) = Re(r(#)f) & r(#)(Im f) = Im(r(#)f)
proof
A1: dom(r(#)(Re f)) = dom Re f by VALUED_1:def 5;
A3: Im r = 0 by COMPLEX1:def 2;
A4: dom Re f = dom f by COMSEQ_3:def 3;
A5: Re r = r by COMPLEX1:def 1;
A6: dom(r(#)f) = dom f by VALUED_1:def 5;
A7: dom Re(r(#)f) = dom(r(#)f) by COMSEQ_3:def 3;
  now
    let x be object;
A8: Re(r * f.x) = Re r * Re(f.x) - Im r * Im(f.x) by COMPLEX1:9;
    assume
A9: x in dom(r(#)(Re f));
    then
A10: (Re f).x = Re(f.x) by A1,COMSEQ_3:def 3;
    Re(r(#)f).x = Re((r(#)f).x) by A1,A6,A7,A4,A9,COMSEQ_3:def 3;
    then Re(r(#)f).x = r * Re(f.x) by A1,A6,A4,A5,A3,A9,A8,VALUED_1:def 5;
    hence (r(#)(Re f)).x = Re(r(#)f).x by A9,A10,VALUED_1:def 5;
  end;
  hence r(#)(Re f) = Re(r(#)f) by A1,A6,A7,A4,FUNCT_1:2;
A11: dom(r(#)(Im f)) = dom Im f by VALUED_1:def 5;
A12: dom Im f = dom f by COMSEQ_3:def 4;
A13: dom Im(r(#)f) = dom(r(#)f) by COMSEQ_3:def 4;
  now
    let x be object;
A14: Im(r * f.x) = Im r * Re(f.x) + Re r * Im(f.x) by COMPLEX1:9;
    assume
A15: x in dom(r(#)Im(f));
    then
A16: (Im f).x = Im(f.x) by A11,COMSEQ_3:def 4;
    Im(r(#)f).x = Im((r(#)f).x) by A11,A6,A13,A12,A15,COMSEQ_3:def 4;
    then Im(r(#)f).x = r * Im(f.x) by A11,A6,A12,A5,A3,A15,A14,VALUED_1:def 5;
    hence (r(#)Im(f)).x = Im(r(#)f).x by A15,A16,VALUED_1:def 5;
  end;
  hence thesis by A11,A6,A13,A12,FUNCT_1:2;
end;
