reserve x for object, X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for complex-valued Function;
reserve r,p for Complex;
reserve r,r1,r2,p for Real;
reserve f,f1,f2 for PartFunc of C,REAL;

theorem
  chi(X,C) (#) chi(Y,C) = chi(X /\ Y,C)
proof
A1: now
    let c such that
    c in dom (chi(X,C) (#) chi(Y,C));
    now
      per cases;
      suppose
A2:     chi(X,C).c * chi(Y,C).c = 0;
        now
          per cases by A2;
          suppose
            chi(X,C).c = 0;
            then not c in X by Th61;
            then not c in X /\ Y by XBOOLE_0:def 4;
            hence chi(X,C).c * chi(Y,C).c = chi(X /\ Y,C).c by A2,Th61;
          end;
          suppose
            chi(Y,C).c = 0;
            then not c in Y by Th61;
            then not c in X /\ Y by XBOOLE_0:def 4;
            hence chi(X,C).c * chi(Y,C).c = chi(X /\ Y,C).c by A2,Th61;
          end;
        end;
        hence chi(X,C).c * chi(Y,C).c = chi(X /\ Y,C).c;
      end;
      suppose
A3:     chi(X,C).c * chi(Y,C).c <> 0;
        then
A4:     chi(Y,C).c <> 0;
        then
A5:     chi(Y,C).c = 1 by Th67;
A6:     c in Y by A4,Th61;
A7:     chi(X,C).c <> 0 by A3;
        then c in X by Th61;
        then
A8:     c in X /\ Y by A6,XBOOLE_0:def 4;
        chi(X,C).c = 1 by A7,Th67;
        hence chi(X,C).c * chi(Y,C).c = chi(X /\ Y,C).c by A5,A8,Th61;
      end;
    end;
    hence (chi(X,C)(#)chi(Y,C)).c = chi(X /\ Y,C).c by VALUED_1:5;
  end;
  dom (chi(X,C) (#) chi(Y,C)) = dom chi(X,C) /\ dom chi(Y,C) by VALUED_1:def 4
    .= C /\ dom chi(Y,C) by Th61
    .= C /\ C by Th61
    .= dom chi(X /\ Y,C) by Th61;
  hence thesis by A1,PARTFUN1:5;
end;
