
theorem Th63:
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
  E be Element of sigma measurable_rectangles(S1,S2),
  x be Element of X1, y be Element of X2
holds
  ProjPMap1(chi(E,[:X1,X2:]),x) = chi(Measurable-X-section(E,x),X2)
& ProjPMap2(chi(E,[:X1,X2:]),y) = chi(Measurable-Y-section(E,y),X1)
proof
    let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
    A be Element of sigma measurable_rectangles(S1,S2),
    x be Element of X1, y be Element of X2;
    reconsider XX1 = X1 as Element of S1 by MEASURE1:7;
    reconsider XX2 = X2 as Element of S2 by MEASURE1:7;
    reconsider XX12 = [:X1,X2:]
      as Element of sigma measurable_rectangles(S1,S2) by MEASURE1:7;
A1: x in XX1 implies X-section([:XX1,XX2:],x) = XX2 by MEASUR11:22;
    dom(ProjPMap1(chi(A,[:X1,X2:]),x))
     = X-section(dom(chi(A,[:X1,X2:])),x) by Def3
    .= X-section(XX12,x) by FUNCT_3:def 3; then
A2: dom(ProjPMap1(chi(A,[:X1,X2:]),x))
     = dom(chi(Measurable-X-section(A,x),X2)) by A1,FUNCT_3:def 3;

    now let y be Element of X2;
     assume y in dom(ProjPMap1(chi(A,[:X1,X2:]),x));
A3:  [x,y] in [:X1,X2:] by ZFMISC_1:def 2; then
     [x,y] in dom(chi(A,[:X1,X2:])) by FUNCT_3:def 3; then
A4:  ProjPMap1(chi(A,[:X1,X2:]),x).y = chi(A,[:X1,X2:]).(x,y) by Def3;

A5:  Measurable-X-section(A,x) = X-section(A,x) by MEASUR11:def 6
      .= {y where y is Element of X2: [x,y] in A} by MEASUR11:def 4;

     per cases;
     suppose A6: [x,y] in A; then
      y in Measurable-X-section(A,x) by A5; then
      chi(Measurable-X-section(A,x),X2).y = 1 by FUNCT_3:def 3;
      hence ProjPMap1(chi(A,[:X1,X2:]),x).y
         = chi(Measurable-X-section(A,x),X2).y by A4,A6,FUNCT_3:def 3;
     end;
     suppose A7: not [x,y] in A;
      now assume y in Measurable-X-section(A,x); then
       ex y1 be Element of X2 st y1= y & [x,y1] in A by A5;
       hence contradiction by A7;
      end; then
      chi(Measurable-X-section(A,x),X2).y = 0 by FUNCT_3:def 3;
      hence ProjPMap1(chi(A,[:X1,X2:]),x).y
         = chi(Measurable-X-section(A,x),X2).y by A3,A4,A7,FUNCT_3:def 3;
     end;
    end;
    hence ProjPMap1(chi(A,[:X1,X2:]),x) = chi(Measurable-X-section(A,x),X2)
      by A2,PARTFUN1:5;

A8: y in XX2 implies Y-section([:XX1,XX2:],y) = XX1 by MEASUR11:22;

    dom(ProjPMap2(chi(A,[:X1,X2:]),y))
     = Y-section(dom(chi(A,[:X1,X2:])),y) by Def4
    .= Y-section(XX12,y) by FUNCT_3:def 3; then
A9: dom(ProjPMap2(chi(A,[:X1,X2:]),y))
     = dom(chi(Measurable-Y-section(A,y),X1)) by A8,FUNCT_3:def 3;

    now let x be Element of X1;
     assume x in dom(ProjPMap2(chi(A,[:X1,X2:]),y));
A10: [x,y] in [:X1,X2:] by ZFMISC_1:def 2; then
     [x,y] in dom(chi(A,[:X1,X2:])) by FUNCT_3:def 3; then
A11: ProjPMap2(chi(A,[:X1,X2:]),y).x = chi(A,[:X1,X2:]).(x,y) by Def4;

A12: Measurable-Y-section(A,y) = Y-section(A,y) by MEASUR11:def 7
      .= {x where x is Element of X1: [x,y] in A} by MEASUR11:def 5;

     per cases;
     suppose A13: [x,y] in A; then
      x in Measurable-Y-section(A,y) by A12; then
      chi(Measurable-Y-section(A,y),X1).x = 1 by FUNCT_3:def 3;
      hence ProjPMap2(chi(A,[:X1,X2:]),y).x
         = chi(Measurable-Y-section(A,y),X1).x by A11,A13,FUNCT_3:def 3;
     end;
     suppose A14: not [x,y] in A;
      now assume x in Measurable-Y-section(A,y); then
       ex x1 be Element of X1 st x1= x & [x1,y] in A by A12;
       hence contradiction by A14;
      end; then
      chi(Measurable-Y-section(A,y),X1).x = 0 by FUNCT_3:def 3;
      hence ProjPMap2(chi(A,[:X1,X2:]),y).x
         = chi(Measurable-Y-section(A,y),X1).x by A10,A11,A14,FUNCT_3:def 3;
     end;
    end;
    hence thesis by A9,PARTFUN1:5;
end;
