
theorem
  for X being set, R being Relation of X
  holds R is antisymmetric iff R \ id X is asymmetric
proof
  let X be set, R be Relation of X;
  per cases;
  suppose A1: X is non empty;
    hereby
      assume R is antisymmetric;
      then R is_antisymmetric_in X by A1, PARTIT_2:24;
      hence R \ id X is asymmetric by A1, RELAT_2:3, PARTIT_2:29;
    end;
    assume R \ id X is asymmetric;
    then R \ id X is_asymmetric_in X by A1, PARTIT_2:28;
    hence R is antisymmetric by A1, RELAT_2:3, PARTIT_2:25;
  end;
  suppose X is empty;
    hence thesis;
  end;
end;
