
theorem Th26:
  for X,Y,A,B being set, x,y being object st x in X & y in Y holds
    chi(A,X).x * chi(B,Y).y = chi([:A,B:],[:X,Y:]).(x,y)
proof
   let X,Y,A,B be set, x,y be object;
   assume A1: x in X & y in Y;
   per cases;
   suppose A2: x in A & y in B; then
A3: [x,y] in [:X,Y:] & [x,y] in [:A,B:] by A1,ZFMISC_1:87;
    chi(A,X).x = 1 & chi(B,Y).y = 1 by A1,A2,FUNCT_3:def 3; then
    chi(A,X).x * chi(B,Y).y = 1 * 1 by XXREAL_3:def 5;
    hence chi(A,X).x * chi(B,Y).y = chi([:A,B:],[:X,Y:]).(x,y)
      by A3,FUNCT_3:def 3;
   end;
   suppose A4: not x in A or not y in B; then
    not [x,y] in [:A,B:] & [x,y] in [:X,Y:] by A1,ZFMISC_1:87; then
A5: chi([:A,B:],[:X,Y:]).(x,y) = 0 by FUNCT_3:def 3;
    per cases by A4;
    suppose not x in A; then
     chi(A,X).x = 0 by A1,FUNCT_3:def 3;
     hence chi(A,X).x * chi(B,Y).y = chi([:A,B:],[:X,Y:]).(x,y) by A5;
    end;
    suppose not y in B; then
     chi(B,Y).y = 0 by A1,FUNCT_3:def 3;
     hence chi(A,X).x * chi(B,Y).y = chi([:A,B:],[:X,Y:]).(x,y) by A5;
    end;
   end;
end;
