
theorem Th16:
for X be set, A1,A2 be Subset of X, er be ExtReal, x be object st A1 misses A2
 holds (chi(er,A1,X)|A2).x = 0
proof
   let X be set, A1,A2 be Subset of X, er be ExtReal, x be object;
   assume a1: A1 misses A2;
   per cases;
   suppose a2: x in dom(chi(er,A1,X)|A2); then
    x in dom chi(er,A1,X) /\ A2 by RELAT_1:61; then
    x in X & x in A2 by XBOOLE_0:def 4; then
    not x in A1 by a1,XBOOLE_0:3; then
    chi(er,A1,X).x = 0 by a2,Def1;
    hence (chi(er,A1,X)|A2).x = 0 by a2,FUNCT_1:47;
   end;
   suppose not x in dom(chi(er,A1,X)|A2);
    hence (chi(er,A1,X)|A2).x = 0 by FUNCT_1:def 2;
   end;
end;
