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
  c in C \ X iff chi(X,C).c = 0
proof
  thus c in C \ X implies chi(X,C).c = 0
  proof
    assume c in C \ X;
    then not c in X by XBOOLE_0:def 5;
    hence thesis by Th61;
  end;
  assume chi(X,C).c = 0;
  then not c in X by Th61;
  hence thesis by XBOOLE_0:def 5;
end;
