
theorem
for X,Y be non empty set, x,y be set, E1,E2 be Subset of [:X,Y:]
 st E1 misses E2 holds
   chi(E1 \/ E2,[:X,Y:]).(x,y)
     = chi(X-section(E1,x),Y).y + chi(X-section(E2,x),Y).y
 & chi(E1 \/ E2,[:X,Y:]).(x,y)
     = chi(Y-section(E1,y),X).x + chi(Y-section(E2,y),X).x
proof
   let X,Y be non empty set, x,y be set, E1,E2 be Subset of [:X,Y:];
   assume E1 misses E2; then
A1:X-section(E1,x) misses X-section(E2,x)
 & Y-section(E1,y) misses Y-section(E2,y) by Th29;
A2:chi(E1 \/ E2,[:X,Y:]).(x,y) = chi(X-section(E1 \/ E2,x),Y).y by Th28
    .= chi(X-section(E1,x) \/ X-section(E2,x),Y).y by Th20;
   thus chi(E1 \/ E2,[:X,Y:]).(x,y)
         = chi(X-section(E1,x),Y).y + chi(X-section(E2,x),Y).y
   proof
    per cases;
    suppose B1: not y in Y;
     dom( chi(X-section(E1,x) \/ X-section(E2,x),Y) ) = Y
   & dom( chi(X-section(E1,x),Y) ) = Y
   & dom( chi(X-section(E2,x),Y) ) = Y by FUNCT_3:def 3; then
     chi(E1 \/ E2,[:X,Y:]).(x,y) = 0 &
     chi(X-section(E1,x),Y).y = 0 & chi(X-section(E2,x),Y).y = 0
      by A2,B1,FUNCT_1:def 2;
     hence thesis;
    end;
    suppose A3: y in Y & y in X-section(E1,x) \/ X-section(E2,x); then
A4:  chi(E1 \/ E2,[:X,Y:]).(x,y) = 1 by A2,FUNCT_3:def 3;
     per cases by A1,A3,XBOOLE_0:5;
     suppose y in X-section(E1,x) & not y in X-section(E2,x); then
      chi(X-section(E1,x),Y).y = 1 & chi(X-section(E2,x),Y).y = 0
        by FUNCT_3:def 3;
      hence thesis by A4,XXREAL_3:4;
     end;
     suppose not y in X-section(E1,x) & y in X-section(E2,x); then
      chi(X-section(E1,x),Y).y = 0 & chi(X-section(E2,x),Y).y = 1
        by FUNCT_3:def 3;
      hence thesis by A4,XXREAL_3:4;
     end;
    end;
    suppose A5: y in Y & not y in X-section(E1,x) \/ X-section(E2,x); then
A6:  chi(E1 \/ E2,[:X,Y:]).(x,y) = 0 by A2,FUNCT_3:def 3;
     not y in X-section(E1,x) & not y in X-section(E2,x)
       by A5,XBOOLE_0:def 3; then
     chi(X-section(E1,x),Y).y = 0 & chi(X-section(E2,x),Y).y = 0
       by A5,FUNCT_3:def 3;
     hence thesis by A6;
    end;
   end;
C2:chi(E1 \/ E2,[:X,Y:]).(x,y) = chi(Y-section(E1 \/ E2,y),X).x by Th28
    .= chi(Y-section(E1,y) \/ Y-section(E2,y),X).x by Th20;
    per cases;
    suppose B1: not x in X;
     dom( chi(Y-section(E1,y) \/ Y-section(E2,y),X) ) = X
   & dom( chi(Y-section(E1,y),X) ) = X
   & dom( chi(Y-section(E2,y),X) ) = X by FUNCT_3:def 3; then
     chi(E1 \/ E2,[:X,Y:]).(x,y) = 0 &
     chi(Y-section(E1,y),X).x = 0 & chi(Y-section(E2,y),X).x = 0
      by C2,B1,FUNCT_1:def 2;
     hence thesis;
    end;
    suppose C3: x in X & x in Y-section(E1,y) \/ Y-section(E2,y); then
C4:  chi(E1 \/ E2,[:X,Y:]).(x,y) = 1 by C2,FUNCT_3:def 3;
     per cases by A1,C3,XBOOLE_0:5;
     suppose x in Y-section(E1,y) & not x in Y-section(E2,y); then
      chi(Y-section(E1,y),X).x = 1 & chi(Y-section(E2,y),X).x = 0
        by FUNCT_3:def 3;
      hence thesis by C4,XXREAL_3:4;
     end;
     suppose not x in Y-section(E1,y) & x in Y-section(E2,y); then
      chi(Y-section(E1,y),X).x = 0 & chi(Y-section(E2,y),X).x = 1
        by FUNCT_3:def 3;
      hence thesis by C4,XXREAL_3:4;
     end;
    end;
    suppose C5: x in X & not x in Y-section(E1,y) \/ Y-section(E2,y); then
C6:  chi(E1 \/ E2,[:X,Y:]).(x,y) = 0 by C2,FUNCT_3:def 3;
     not x in Y-section(E1,y) & not x in Y-section(E2,y)
       by C5,XBOOLE_0:def 3; then
     chi(Y-section(E1,y),X).x = 0 & chi(Y-section(E2,y),X).x = 0
       by C5,FUNCT_3:def 3;
     hence thesis by C6;
   end;
end;
