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 Th4:
  -(Im f) = Re(<i>(#)f) & Re f = Im(<i>(#)f)
proof
A1: dom -(Im f) = dom Im f by VALUED_1:8;
A2: dom(<i>(#)f) = dom f by VALUED_1:def 5;
A3: dom Im f = dom f by COMSEQ_3:def 4;
A4: dom Re(<i>(#)f) = dom(<i>(#)f) by COMSEQ_3:def 3;
  now
    let x be object;
A5: (-Im(f)).x = -((Im f).x) by VALUED_1:8;
A6: Re(<i> * f.x) = Re(<i>) * Re(f.x) - Im(<i>) * Im(f.x) by COMPLEX1:9;
    assume
A7: x in dom -(Im f);
    then Re(<i>(#)f).x = Re((<i>(#)f).x) by A1,A4,A2,A3,COMSEQ_3:def 3;
    then Re(<i>(#)f).x = -Im(f.x) by A1,A2,A3,A7,A6,COMPLEX1:7,VALUED_1:def 5;
    hence (-Im(f)).x = Re(<i>(#)f).x by A1,A7,A5,COMSEQ_3:def 4;
  end;
  hence -Im(f) = Re(<i>(#)f) by A4,A2,A3,FUNCT_1:2,VALUED_1:8;
A8: dom Re f = dom f by COMSEQ_3:def 3;
A9: dom Im(<i>(#)f) = dom(<i>(#)f) by COMSEQ_3:def 4;
  now
    let x be object;
A10: Im(<i> * f.x) = Im(<i>) * Re(f.x) + Re(<i>) * Im(f.x) by COMPLEX1:9;
    assume
A11: x in dom Re f;
    then Im(<i>(#)f).x = Im((<i>(#)f).x) by A8,A2,A9,COMSEQ_3:def 4;
    then Im(<i>(#)f).x = Re(f.x) by A8,A2,A11,A10,COMPLEX1:7,VALUED_1:def 5;
    hence Re(f).x = Im(<i>(#)f).x by A11,COMSEQ_3:def 3;
  end;
  hence thesis by A8,A2,A9,FUNCT_1:2;
end;
