
theorem Th14:
for X be set, A1,A2 be Subset of X, er be ExtReal holds
 chi(er,A1,X)|A2 = chi(er,A1/\A2,X)|A2
proof
   let X be set, A1,A2 be Subset of X, er be ExtReal;
a1:dom(chi(er,A1,X)|A2) = dom chi(er,A1,X) /\ A2 by RELAT_1:61
    .= X /\ A2 by FUNCT_2:def 1;
a2:dom(chi(er,A1/\A2,X)|A2) = dom chi(er,A1/\A2,X) /\ A2 by RELAT_1:61
    .= dom(chi(er,A1,X)|A2) by a1,FUNCT_2:def 1;
   now let x be Element of X;
    assume b1: x in dom(chi(er,A1,X)|A2); then
a3: x in X & x in A2 by a1,XBOOLE_0:def 4; then
a4: (chi(er,A1,X)|A2).x = chi(er,A1,X).x
  & (chi(er,A1/\A2,X)|A2).x = chi(er,A1/\A2,X).x by FUNCT_1:49;
    per cases;
    suppose a5: x in A1; then
a6:  (chi(er,A1,X)|A2).x = er by a4,Def1;
     x in A1 /\ A2 by a3,a5,XBOOLE_0:def 4;
     hence (chi(er,A1/\A2,X)|A2).x = (chi(er,A1,X)|A2).x by a4,a6,Def1;
    end;
    suppose a7: not x in A1; then
a8:  (chi(er,A1,X)|A2).x = 0 by a4,Def1,b1;
     not x in A1 /\ A2 by a7,XBOOLE_0:def 4;
     hence (chi(er,A1/\A2,X)|A2).x = (chi(er,A1,X)|A2).x
       by b1,a4,a8,Def1;
    end;
   end;
   hence chi(er,A1,X)|A2 = chi(er,A1/\A2,X)|A2 by a2,PARTFUN1:5;
end;
