reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,ExtREAL,
  E for Element of S;

theorem Th15:
  dom f /\ dom g = E & f is real-valued & g is real-valued & f
  is E-measurable & g is E-measurable implies f(#)g is E-measurable
proof
  assume that
A1: dom f /\ dom g = E and
A2: f is real-valued and
A3: g is real-valued and
A4: f is E-measurable and
A5: g is E-measurable;
A6: dom (f(#)g) = dom f /\ dom g by MESFUNC1:def 5;
A7: dom ((1/4)(#)(|.f+g.|)|^2) = dom ((|.f+g.|)|^2) by MESFUNC1:def 6;
A8: dom ((|.f-g.|)|^2) = dom |.f-g.| by Def4;
  then
A9: dom ((|.f-g.|)|^2) = dom (f-g) by MESFUNC1:def 10;
  then
A10: dom ((|.f-g.|)|^2) = dom f /\ dom g by A2,MESFUNC2:2;
  then
A11: dom ((|.f-g.|)|^2) c= dom g by XBOOLE_1:17;
A12: dom ((1/4)(#)(|.f-g.|)|^2) = dom ((|.f-g.|)|^2) by MESFUNC1:def 6;
A13: dom ((|.f+g.|)|^2) = dom |.f+g.| by Def4;
  then
A14: dom ((|.f+g.|)|^2) = dom (f+g) by MESFUNC1:def 10;
  then
A15: dom ((|.f+g.|)|^2) = dom f /\ dom g by A2,MESFUNC2:2;
  then
A16: dom ((|.f+g.|)|^2) c= dom g by XBOOLE_1:17;
A17: dom ((|.f+g.|)|^2) c= dom f by A15,XBOOLE_1:17;
  for x be Element of X st x in dom ((|.f+g.|)|^2)
   holds |. ((|.f+g.|)|^2).x .| < +infty
  proof
    let x be Element of X;
    assume
A18: x in dom ((|. f+g .|)|^2);
    then
A19: |.g.x.| < +infty by A3,A16,MESFUNC2:def 1;
    |.f.x.| < +infty by A2,A17,A18,MESFUNC2:def 1;
    then reconsider c1 = f.x, c2 = g.x as Element of REAL by A19,EXTREAL1:41;
    f.x + g.x = c1 + c2 by SUPINF_2:1;
    then |. f.x + g.x .| = |.c1+c2 qua Complex.| by EXTREAL1:12;
    then
A20: |. f.x + g.x .| * |. f.x + g.x .|
         = |.c1+c2 qua Complex.|*|.c1+c2 qua Complex.| by EXTREAL1:1;
    ((|.f+g.|)|^2).x = ((|.f+g.|).x)|^(1+1) by A18,Def4;
    then
A21: ((|.f+g.|)|^2).x = (|.f+g.| .x)|^1 * (|.f+g.| .x) by Th10;
A22: |.(f+g).x.| = |. f.x+g.x .| by A14,A18,MESFUNC1:def 3;
    |.f+g.| .x = |.(f+g).x.| by A13,A18,MESFUNC1:def 10;
    then
    |.((|.f+g.|)|^2).x.| = |. |. f.x + g.x .| * |. f.x + g.x .| .| by A21,A22
,Th9
      .= |. |.(f.x + g.x) * (f.x + g.x).| .| by EXTREAL1:19
      .= |.(f.x + g.x) * (f.x + g.x).|
      .= |. f.x + g.x .| * |. f.x + g.x .| by EXTREAL1:19;
    hence thesis by A20,XXREAL_0:9, XREAL_0:def 1;
  end;
  then (|.f+g.|)|^2 is real-valued by MESFUNC2:def 1;
  then
A23: (1/4)(#)(|.f+g.|)|^2 is real-valued by MESFUNC2:10;
  then
A24: dom ((1/4)(#)(|.f+g.|)|^2 - (1/4)(#)(|.f-g.|)|^2) = dom((1/4)(#)(|.f+g
  .|)|^2) /\ dom((1/4)(#)(|.f-g.|)|^2) by MESFUNC2:2;
  for x be Element of X st x in dom (f(#)g) holds (f(#)g).x = ((1/4)(#)(
  |.f+g.|) |^ 2 - (1/4)(#)(|.f-g.|) |^ 2).x
  proof
    let x be Element of X;
    assume
A25: x in dom (f(#)g);
    then
A26: |.g.x.| <+infty by A3,A15,A16,A6,MESFUNC2:def 1;
    |.f.x.| <+infty by A2,A15,A17,A6,A25,MESFUNC2:def 1;
    then reconsider c1 = f.x, c2 = g.x as Element of REAL by A26,EXTREAL1:41;
    f.x + g.x = c1 + c2 by SUPINF_2:1;
    then |. f.x+g.x .| = |.c1+c2 qua Complex.| by EXTREAL1:12;
    then
A27: |. f.x+g.x .| * |. f.x+g.x .|
      = |.c1+c2 qua Complex.|*|.c1+c2 qua Complex.| by EXTREAL1:1;
    ((1/4)(#)(|.f+g.|)|^2).x = (1/4) * ((|.f+g.|)|^2).x by A15,A6,A7,A25,
MESFUNC1:def 6;
    then ((1/4)(#)(|.f+g.|)|^2).x = (1/4) * (|.f+g.| .x)|^(1+1) by A15,A6
,A25,Def4;
    then
A28: ((1/4)(#)(|.f+g.|)|^2).x = (1/4) * ((|.f+g.| .x)|^1 * (|.f+g.| .
    x) ) by Th10;
A29: |.f+g.| .x = |.(f+g).x.| by A13,A15,A6,A25,MESFUNC1:def 10;
    |.(f+g).x.| = |. f.x+g.x .| by A14,A15,A6,A25,MESFUNC1:def 3;
    then ((1/4)(#)(|.f+g.|)|^2).x = (1/4) * (|. f.x+g.x .| * |. f. x+g.x
    .| ) by A29,A28,Th9;
    then
A30: ((1/4)(#)(|.f+g.|)|^2).x
     =(1/4)*(|.c1+c2 qua Complex.|*|.c1+c2 qua Complex.|) by A27,EXTREAL1:1;
    (|.c1-c2 qua Complex.|*|.c1-c2 qua Complex.|) = (|.c1-c2 qua Complex.|)^2;
    then
A31: (|.c1-c2 qua Complex.|*|.c1-c2 qua Complex.|) = (c1-c2)^2 by COMPLEX1:75;
    ((1/4)(#)(|.f-g.|)|^2).x = (1/4) * ((|.f-g.|)|^2).x by A10,A6,A12,A25,
MESFUNC1:def 6;
    then ((1/4)(#)(|.f-g.|)|^2).x = (1/4) * (|.f-g.| .x)|^(1+1) by A10,A6
,A25,Def4;
    then
A32: ((1/4)(#)(|.f-g.|)|^2).x = (1/4) * ((|.f-g.| .x)|^1 * (|.f-g.| .
    x) ) by Th10;
    f.x - g.x = c1 - c2 by SUPINF_2:3;
    then |. f.x-g.x .| = |.c1-c2 qua Complex.| by EXTREAL1:12;
    then
A33: |. f.x-g.x .| * |. f.x-g.x .|
          = |.c1-c2 qua Complex.|*|.c1-c2 qua Complex.| by EXTREAL1:1;
A34: |.f-g.| .x = |.(f-g).x.| by A8,A10,A6,A25,MESFUNC1:def 10;
    |.(f-g).x.| = |. f.x-g.x .| by A9,A10,A6,A25,MESFUNC1:def 4;
    then ((1/4)(#)(|.f-g.|)|^2).x = (1/4) * (|. f.x-g.x .| * |. f.x-g.x
    .|) by A34,A32,Th9;
    then
A35: ((1/4)(#)(|.f-g.|)|^2).x
       =(1/4)*(|.c1-c2 qua Complex.|*|.c1-c2 qua Complex.|) by A33,EXTREAL1:1;
    (|.c1+c2 qua Complex.|*|.c1+c2 qua Complex.|) = (|.c1+c2 qua Complex.|)^2;
    then (|.c1+c2 qua Complex.|*|.c1+c2 qua Complex.|) = (c1+c2)^2
              by COMPLEX1:75;
    then
    ((1/4)(#)(|.f+g.|)|^2).x -((1/4)(#)(|.f-g.|)|^2).x =(1/4)*(c1^2+2*c1*
    c2+c2^2) - (1/4)*(c1^2-2*c1*c2+c2^2) by A30,A35,A31,SUPINF_2:3
      .= c1*c2
      .= (f.x)*(g.x) by EXTREAL1:1
      .= (f(#)g).x by A25,MESFUNC1:def 5;
    hence thesis by A15,A10,A6,A7,A12,A24,A25,MESFUNC1:def 4;
  end;
  then
A36: f(#)g = (1/4)(#)(|.f+g.|)|^2 - (1/4)(#)(|.f-g.|)|^2 by A15,A10,A6,A7,A12
,A24,PARTFUN1:5;
A37: dom ((|.f-g.|)|^2) c= dom f by A10,XBOOLE_1:17;
  for x be Element of X st x in dom ((|.f-g.|)|^2) holds |.((|.f-g.|)|^2)
  .x .| < +infty
  proof
    let x be Element of X;
    assume
A38: x in dom ((|.f-g.|)|^2);
    then
A39: |.g.x.| < +infty by A3,A11,MESFUNC2:def 1;
    |.f.x.| < +infty by A2,A37,A38,MESFUNC2:def 1;
    then reconsider c1 = f.x, c2 = g.x as Element of REAL by A39,EXTREAL1:41;
    f.x - g.x = c1 - c2 by SUPINF_2:3;
    then |. f.x-g.x.| = |.c1-c2 qua Complex.| by EXTREAL1:12;
    then
A40: |. f.x-g.x .| * |. f.x-g.x .|
       = |.c1-c2 qua Complex.|*|.c1-c2 qua Complex.| by EXTREAL1:1;
    ((|.f-g.|)|^2).x = ((|.f-g.|).x)|^(1+1) by A38,Def4;
    then
A41: ((|.f-g.|)|^2).x = ((|.f-g.|).x)|^1 * (|.f-g.|).x by Th10;
    ((|.f-g.|).x) = |.(f-g).x.| by A8,A38,MESFUNC1:def 10;
    then ((|.f-g.|).x) = |. f.x-g.x .| by A9,A38,MESFUNC1:def 4;
    then
    |.((|.f-g.|)|^2).x.| = |. |. f.x - g.x .| * |. f.x - g.x .| .| by A41,Th9
      .= |. |.(f.x - g.x) * (f.x - g.x).| .| by EXTREAL1:19
      .= |.(f.x - g.x) * (f.x - g.x).|
      .= |. f.x - g.x .| * |. f.x - g.x .| by EXTREAL1:19;
    hence thesis by A40,XXREAL_0:9,XREAL_0:def 1;
  end;
  then (|.f-g.|)|^2 is real-valued by MESFUNC2:def 1;
  then
A42: (1/4)(#)(|.f-g.|)|^2 is real-valued by MESFUNC2:10;
  f+g is E-measurable by A2,A3,A4,A5,MESFUNC2:7;
  then (|.f+g.|)|^2 is E-measurable by A1,A14,A15,Th14;
  then
A43: (jj/4)(#)(|.f+g.|)|^2 is E-measurable by A1,A15,MESFUNC5:49;
  f-g is E-measurable by A1,A2,A3,A4,A5,MESFUNC2:11,XBOOLE_1:17;
  then (|.f-g.|)|^2 is E-measurable by A1,A9,A10,Th14;
  then (jj/4)(#)(|.f-g.|)|^2 is E-measurable by A1,A10,MESFUNC5:49;
  hence thesis by A1,A10,A12,A23,A42,A36,A43,MESFUNC2:11;
end;
