
theorem
  for M being non empty MetrSpace, V being Subset of M holds (V is
  equivalence_class of M iff ex x being Element of M st V={x})
proof
  let M be non empty MetrSpace, V be Subset of M;
A1: V is equivalence_class of M implies ex x being Element of M st V={x}
  proof
    assume V is equivalence_class of M;
    then consider x being Element of M such that
A2: V=x -neighbour by Def3;
    x-neighbour={x} by Th13;
    hence thesis by A2;
  end;
  (ex x being Element of M st V={x}) implies V is equivalence_class of M
  proof
    given x being Element of M such that
A3: V={x};
    {x}=x-neighbour by Th13;
    hence thesis by A3,Def3;
  end;
  hence thesis by A1;
end;
