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

theorem Th50:
  for y being object
  for f1,f2 being Function of X,{y} holds f1 = f2
proof let y be object;
  let f1,f2 be Function of X,{y};
  for x being object holds x in X implies f1.x = f2.x
  proof let x be object;
    assume
A1: x in X;
    then f1.x = y by Th49;
    hence thesis by A1,Th49;
  end;
  hence thesis by Th12;
end;
