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 Th30:
  f is A-measurable & A c= dom f implies |.f.| is A-measurable
proof
  assume that
A1: f is A-measurable and
A2: A c= dom f;
A3: dom Im f = dom f by COMSEQ_3:def 4;
A4: now
    let x be object;
    assume x in dom |.Im f.|;
     then |.Im f.|.x = |.(Im f).x qua Complex .| by VALUED_1:def 11;
    hence 0 <= |.Im f.|.x by COMPLEX1:46;
  end;
  then
A5: |.Im f.| to_power 2 is nonnegative by Th26,MESFUNC6:52;
A6: dom |.Im f.| = dom Im f by VALUED_1:def 11;
  then
A7: dom(|.Im f.| to_power 2) = dom f by A3,Def4;
  Im f is A-measurable by A1;
  then |.Im f.| is A-measurable by A2,A3,MESFUNC6:48;
  then
A8: |.Im f.| to_power 2 is A-measurable by A2,A6,A3,A4,Th29,MESFUNC6:52;
A9: dom Re f = dom f by COMSEQ_3:def 3;
A10: now
    let x be object;
A11: 0 <= |.(Re f).x qua Complex .| by COMPLEX1:46;
    assume x in dom |.Re f.|;
    hence 0 <= |. Re f.|.x by A11,VALUED_1:def 11;
  end;
  then
A12: |.Re(f).| to_power 2 is nonnegative by Th26,MESFUNC6:52;
  set F = |.Re f.| to_power 2 + |.Im f.| to_power 2;
A13: dom |.f.| = dom f by VALUED_1:def 11;
A14: dom |.Re f.| = dom Re f by VALUED_1:def 11;
  then
A15: dom(|.Re f.| to_power 2) = dom f by A9,Def4;
A16: dom F = dom( |.Re f.| to_power 2 ) /\ dom( |.Im f.| to_power 2 ) by
VALUED_1:def 1;
  then
A17: dom(|.f.|) = dom (F to_power (1/2)) by A13,A15,A7,Def4;
  now
    let x be object;
    assume
A18: x in dom |.f.|;
    then (|.Re f.| to_power 2).x = (|.Re f.|.x) to_power 2 by A13,A15,Def4;
    then (|.Re f.| to_power 2).x
        = |.Re(f).x qua Complex .| to_power 2 by A14,A13,A9,A18,
VALUED_1:def 11;
    then (|.Re f.| to_power 2).x =
      |.Re(f.x) qua Complex.| to_power 2 by A13,A9,A18,
COMSEQ_3:def 3;
    then (|.Re f.| to_power 2).x = |.Re(f.x) qua Complex.|^2 by POWER:46;
    then
A19: (|.Re f.| to_power 2).x = (Re(f.x))^2 by COMPLEX1:75;
    (|.Im f.| to_power 2).x = (|.Im f.|.x) to_power 2 by A13,A7,A18,Def4;
    then (|.Im f.| to_power 2).x
          = |.Im(f).x qua Complex .| to_power 2 by A6,A13,A3,A18,
VALUED_1:def 11;
    then (|.Im f.| to_power 2).x
          = |.Im(f.x) qua Complex.| to_power 2 by A13,A3,A18,
COMSEQ_3:def 4;
    then (|.Im f.| to_power 2).x = |.Im(f.x) qua Complex.|^2 by POWER:46;
    then
A20: (|.Im f.| to_power 2).x = (Im(f.x))^2 by COMPLEX1:75;
    (F.x) to_power (1/2) = sqrt (F.x) by A12,A5,Th27,MESFUNC6:56
      .= sqrt ( (Re(f.x))^2 + (Im(f.x))^2 ) by A13,A16,A15,A7,A18,A19,A20,
VALUED_1:def 1;
    then (F to_power (1/2)).x = |.f.x .| by A17,A18,Def4;
    hence |.f.|.x = (F to_power (1/2)).x by A18,VALUED_1:def 11;
  end;
  then
A21: |.f.| = F to_power (1/2) by A17,FUNCT_1:2;
  Re f is A-measurable by A1;
  then |.Re f.| is A-measurable by A2,A9,MESFUNC6:48;
  then |.Re f.| to_power 2 is A-measurable by A2,A14,A9,A10,Th29,MESFUNC6:52
;
  then F is A-measurable by A8,MESFUNC6:26;
  hence thesis by A2,A12,A5,A16,A15,A7,A21,Th29,MESFUNC6:56;
end;
