
theorem Th48:
for X1,X2 be non empty set, E be Subset of [:X1,X2:],
 x be Element of X1, y be Element of X2 holds
   ProjPMap1(chi(E,[:X1,X2:]),x) = chi(X-section(E,x),X2) &
   ProjPMap2(chi(E,[:X1,X2:]),y) = chi(Y-section(E,y),X1)
proof
   let X1,X2 be non empty set, E be Subset of [:X1,X2:],
   x be Element of X1, y be Element of X2;

   for y be Element of X2 holds
     ProjMap1(chi(E,[:X1,X2:]),x).y = chi(X-section(E,x),X2).y
   proof
    let y be Element of X2;
A1:ProjMap1(chi(E,[:X1,X2:]),x).y = chi(E,[:X1,X2:]).(x,y)
     by MESFUNC9:def 6; then
A2:[x,y] in E implies ProjMap1(chi(E,[:X1,X2:]),x).y = 1 by FUNCT_3:def 3;

   [x,y] is Element of [:X1,X2:] by ZFMISC_1:def 2; then
A3:not [x,y] in E implies ProjMap1(chi(E,[:X1,X2:]),x).y = 0
      by A1,FUNCT_3:def 3;
    per cases;
    suppose A4: [x,y] in E; then
     y in X-section(E,x) by Th25;
     hence ProjMap1(chi(E,[:X1,X2:]),x).y = chi(X-section(E,x),X2).y
      by A2,A4,FUNCT_3:def 3;
    end;
    suppose A5: not [x,y] in E; then
     not y in X-section(E,x) by Th25;
     hence ProjMap1(chi(E,[:X1,X2:]),x).y = chi(X-section(E,x),X2).y
       by A3,A5,FUNCT_3:def 3;
    end;
   end; then
   ProjMap1(chi(E,[:X1,X2:]),x)=chi(X-section(E,x),X2) by FUNCT_2:def 8;
   hence ProjPMap1(chi(E,[:X1,X2:]),x)=chi(X-section(E,x),X2) by Th27;

   for x be Element of X1 holds
     ProjMap2(chi(E,[:X1,X2:]),y).x = chi(Y-section(E,y),X1).x
   proof
    let x be Element of X1;
A1:ProjMap2(chi(E,[:X1,X2:]),y).x = chi(E,[:X1,X2:]).(x,y)
     by MESFUNC9:def 7; then
A2:[x,y] in E implies ProjMap2(chi(E,[:X1,X2:]),y).x = 1 by FUNCT_3:def 3;

   [x,y] is Element of [:X1,X2:] by ZFMISC_1:def 2; then
A3:not [x,y] in E implies ProjMap2(chi(E,[:X1,X2:]),y).x = 0
      by A1,FUNCT_3:def 3;
    per cases;
    suppose A4: [x,y] in E; then
     x in Y-section(E,y) by Th25;
     hence ProjMap2(chi(E,[:X1,X2:]),y).x = chi(Y-section(E,y),X1).x
       by A2,A4,FUNCT_3:def 3;
    end;
    suppose A5: not [x,y] in E; then
     not x in Y-section(E,y) by Th25;
     hence ProjMap2(chi(E,[:X1,X2:]),y).x = chi(Y-section(E,y),X1).x
       by A3,A5,FUNCT_3:def 3;
    end;
   end; then
   ProjMap2(chi(E,[:X1,X2:]),y)=chi(Y-section(E,y),X1) by FUNCT_2:def 8;
   hence ProjPMap2(chi(E,[:X1,X2:]),y)=chi(Y-section(E,y),X1) by Th27;
end;
