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 Th31:
  (ex A be Element of S st A = dom f & f is A-measurable )
  implies (f is_integrable_on M iff |.f.| is_integrable_on M)
proof
A1: dom (|.Re f.| + |.Im f.|) = dom |.Re f.| /\ dom |.Im f.| by VALUED_1:def 1;
A2: dom |.Re f.| = dom Re f by VALUED_1:def 11;
A3: dom |.Im f.| = dom Im f by VALUED_1:def 11;
A4: dom |.f.| = dom f by VALUED_1:def 11;
  assume ex A be Element of S st A = dom f & f is A-measurable;
  then consider A be Element of S such that
A5: A = dom f and
A6: f is A-measurable;
A7: dom Re f = A by A5,COMSEQ_3:def 3;
A8: Im f is A-measurable by A6;
A9: Re f is A-measurable by A6;
A10: dom Im f = A by A5,COMSEQ_3:def 4;
A11: |.f.| is A-measurable by A5,A6,Th30;
  hereby
A12: now
      let x be Element of X;
      assume
A13:  x in dom |.f.|;
      then
A14:  |.f.|.x = |.f.x .| by VALUED_1:def 11;
A15:  |.(Im f).x qua Complex .| = |.Im f.|.x
          by A5,A10,A3,A4,A13,VALUED_1:def 11;
A16:  |.(Re f).x qua Complex .| = |.Re f.|.x
           by A5,A7,A2,A4,A13,VALUED_1:def 11;
A17:  (Im f).x = Im(f.x) by A5,A10,A4,A13,COMSEQ_3:def 4;
A18:  (Re f).x = Re(f.x) by A5,A7,A4,A13,COMSEQ_3:def 3;
      (|.Re f.| + |.Im f.|).x = |.Re f.|.x + |.Im f.|.x by A5,A7,A10,A1,A2,A3
,A4,A13,VALUED_1:def 1;
      hence |.(|.f.|.x  qua Complex).|
     <= (|.Re f.| + |.Im f.|).x by A18,A17,A14,A16,A15,
COMPLEX1:78;
    end;
    assume
A19: f is_integrable_on M;
    then Im f is_integrable_on M;
    then
A20: |.Im f.| is_integrable_on M by A10,A8,MESFUNC6:94;
    Re f is_integrable_on M by A19;
    then |.Re f.| is_integrable_on M by A7,A9,MESFUNC6:94;
    then |.Re f.| + |.Im f.| is_integrable_on M by A20,MESFUNC6:100;
    hence |.f.| is_integrable_on M by A5,A7,A10,A1,A2,A3,A4,A11,A12,MESFUNC6:96
;
  end;
  hereby
    assume
A21: |.f.| is_integrable_on M;
    now
      let x be Element of X;
A22:  |.Im(f.x) qua Complex.| <= |.f.x.| by COMPLEX1:79;
      assume
A23:  x in dom Im f;
      then |.f.|.x = |.f.x .| by A5,A10,A4,VALUED_1:def 11;
      hence |.Im(f).x qua Complex .| <= |.f.|.x
                by A23,A22,COMSEQ_3:def 4;
    end;
    then
A24: Im f is_integrable_on M by A5,A10,A8,A4,A21,MESFUNC6:96;
    now
      let x be Element of X;
A25:  |.Re(f.x) qua Complex.| <= |.f.x .| by COMPLEX1:79;
      assume
A26:  x in dom Re f;
      then |.f.|.x = |.f.x .| by A5,A7,A4,VALUED_1:def 11;
      hence |.Re(f).x qua Complex .| <= |.f.|.x
            by A26,A25,COMSEQ_3:def 3;
    end;
    then Re f is_integrable_on M by A5,A7,A9,A4,A21,MESFUNC6:96;
    hence f is_integrable_on M by A24;
  end;
end;
