 reserve n for Nat;

theorem ThEgal:
  for a, b being POINT of TarskiEuclid2Space holds a = b iff dist(a,b) = 0
  proof
    let a, b be POINT of TarskiEuclid2Space;
    hereby
      assume a = b;
      then Tn2R a = Tn2R b;
      then 0 = |. Tn2TR a - Tn2TR b .|;
      hence dist(a,b) = 0 by ThEquiv;
    end;
    assume dist(a,b) = 0;
    hence thesis by METRIC_1:2;
  end;
