reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;
reserve D for non empty set;
reserve A,B for non empty set;
reserve Y for non empty set,
  f for Function of X,Y,
  p for PartFunc of Y,Z,
  x for Element of X;
reserve g for Function of X,X;
reserve X,Y for non empty set,
  Z,S,T for set,
  f for Function of X,Y,
  g for PartFunc of Y,Z,
  x for Element of X;

theorem
  for x being set, A being set
   for f,g being Function of {x}, A st f.x = g.x
   holds f = g
proof
  let x be set, A be set;
  let f,g be Function of {x}, A such that
A1: f.x = g.x;
  now
    let y be object;
    assume y in {x};
    then y = x by TARSKI:def 1;
    hence f.y = g.y by A1;
  end;
  hence thesis;
end;
