reserve a, b for Real,
  r for Real,
  rr for Real,
  i, j, n for Nat,
  M for non empty MetrSpace,
  p, q, s for Point of TOP-REAL 2,
  e for Point of Euclid 2,
  w for Point of Euclid n,
  z for Point of M,
  A, B for Subset of TOP-REAL n,
  P for Subset of TOP-REAL 2,
  D for non empty Subset of TOP-REAL 2;
reserve a, b for Real;
reserve a, b for Real;
reserve r for Real;
reserve r1,r2,s1,s2 for Real;

theorem
  for p, q, r being Point of TOP-REAL n holds dist(p,r) <= dist (p,q) +
  dist(q,r)
proof
  let p, q, r be Point of TOP-REAL n;
A1: ex a, b being Point of Euclid n st a = p & b = r & dist(a,b) = dist(p,r
  ) by Def1;
A2: ex a, b being Point of Euclid n st a = q & b = r & dist(a,b) = dist(q,r)
  by Def1;
  ex a, b being Point of Euclid n st a = p & b = q & dist(a,b) = dist(p,q
  ) by Def1;
  hence thesis by A1,A2,METRIC_1:4;
end;
