 reserve n for Nat;

theorem ThNull:
  for a,b,c,d being POINT of TarskiEuclid2Space st
    dist(a,b) + dist(c,d) = 0 holds a = b & c = d
  proof
    let a,b,c,d be POINT of TarskiEuclid2Space;
    assume
A1: dist(a,b) + dist(c,d) = 0;
    0 <= dist(a,b) & 0 <= dist(c,d) by METRIC_1:5;
    then dist(a,b) = 0 & dist(c,d) = 0 by A1;
    hence thesis by ThEgal;
  end;
