
theorem Th35:
for X1,X2 be non empty set, A be Subset of [:X1,X2:], x be Element of X1,
  y be Element of X2 holds
   ProjPMap1(Xchi(A,[:X1,X2:]),x) = Xchi(X-section(A,x),X2)
 & ProjPMap2(Xchi(A,[:X1,X2:]),y) = Xchi(Y-section(A,y),X1)
proof
   let X1,X2 be non empty set, A be Subset of [:X1,X2:],
   x be Element of X1, y be Element of X2;

A3:ProjPMap1(Xchi(A,[:X1,X2:]),x) = ProjMap1(Xchi(A,[:X1,X2:]),x)
 & ProjPMap2(Xchi(A,[:X1,X2:]),y) = ProjMap2(Xchi(A,[:X1,X2:]),y) by Th27;

   for y be Element of X2 holds
    ProjMap1(Xchi(A,[:X1,X2:]),x).y = Xchi(X-section(A,x),X2).y
   proof
    let y be Element of X2;
a5: [x,y] in [:X1,X2:] by ZFMISC_1:def 2;
a4: ProjMap1(Xchi(A,[:X1,X2:]),x).y = Xchi(A,[:X1,X2:]).(x,y)
      by MESFUNC9:def 6;
    per cases;
    suppose b1: [x,y] in A; then
     y in X-section(A,x) by Th25; then
     ProjMap1(Xchi(A,[:X1,X2:]),x).y = +infty
   & Xchi(X-section(A,x),X2).y = +infty by a4,b1,MEASUR10:def 7;
     hence thesis;
    end;
    suppose b2: not [x,y] in A; then
     not y in X-section(A,x) by Th25; then
     ProjMap1(Xchi(A,[:X1,X2:]),x).y = 0
   & Xchi(X-section(A,x),X2).y = 0 by a5,a4,b2,MEASUR10:def 7;
     hence thesis;
    end;
   end;
   hence ProjPMap1(Xchi(A,[:X1,X2:]),x) = Xchi(X-section(A,x),X2)
     by A3,FUNCT_2:def 8;

   for x be Element of X1 holds
    ProjMap2(Xchi(A,[:X1,X2:]),y).x = Xchi(Y-section(A,y),X1).x
   proof
    let x be Element of X1;
a5: [x,y] in [:X1,X2:] by ZFMISC_1:def 2;
a4: ProjMap2(Xchi(A,[:X1,X2:]),y).x = Xchi(A,[:X1,X2:]).(x,y)
      by MESFUNC9:def 7;
    per cases;
    suppose b1: [x,y] in A; then
     x in Y-section(A,y) by Th25; then
     ProjMap2(Xchi(A,[:X1,X2:]),y).x = +infty
   & Xchi(Y-section(A,y),X1).x = +infty by a4,b1,MEASUR10:def 7;
     hence thesis;
    end;
    suppose b2: not [x,y] in A; then
     not x in Y-section(A,y) by Th25; then
     ProjMap2(Xchi(A,[:X1,X2:]),y).x = 0
   & Xchi(Y-section(A,y),X1).x = 0 by a5,a4,b2,MEASUR10:def 7;
     hence thesis;
    end;
   end;
   hence ProjPMap2(Xchi(A,[:X1,X2:]),y) = Xchi(Y-section(A,y),X1)
     by A3,FUNCT_2:def 8;
end;
