 reserve n for Nat;

theorem ThConv9:
  for a,b,c,d being POINT of TarskiEuclid2Space holds
    dist(a,b)^2 = dist(c,d)^2 iff a,b equiv c,d
  proof
    let a,b,c,d be POINT of TarskiEuclid2Space;
    hereby
      assume
A1:   dist(a,b)^2 = dist(c,d)^2;
      sqrt dist(a,b)^2 = dist(a,b) & sqrt dist(c,d)^2 = dist(c,d)
        by METRIC_1:5,SQUARE_1:22;
      hence a,b equiv c,d by A1,GTARSKI1:def 15;
    end;
    assume a,b equiv c,d;
    hence thesis by GTARSKI1:def 15;
  end;
