reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;

theorem Th12:
  for f1,f2 being Function of X,Y st
  for x being object st x in X holds f1.x = f2.x
  holds f1 = f2
proof
  let f1,f2 be Function of X,Y;
  per cases;
  suppose Y = {};
    hence thesis;
  end;
  suppose Y <> {};
    then dom f1 = X & dom f2 = X by Def1;
    hence thesis;
  end;
end;
