reserve i, n for Nat,
  x, y, a for Real,
  v for Element of n-tuples_on REAL,
  p, p1, p2, p3, q, q1, q2 for Point of TOP-REAL n;

theorem
  |(p,q)| <= |(p,p)| + |(q,q)|
proof
  0 <= |(p,p)| & 0 <= |(q,q)| by Th33;
  then 0 + 0 <= |(p,p)| + |(q,q)| by XREAL_1:7;
  then
A1: 0/2 <= (|(p,p)| + |(q,q)|)/2 by XREAL_1:72;
  |(p-q, p-q)| = |(p,p)| - 2*|(p,q)| + |(q,q)| by Th29
    .= |(p,p)| + |(q,q)| - 2*|(p,q)|;
  then 2*|(p,q)| <= |(p,p)| + |(q,q)| - 0 by Th33,XREAL_1:11;
  then (2*|(p,q)|)/2 <= (|(p,p)| + |(q,q)|)/2 by XREAL_1:72;
  then 0 + |(p,q)| <= (|(p,p)| + |(q,q)|)/2 + (|(p,p)| + |(q,q)|)/2 by A1,
XREAL_1:7;
  hence thesis;
end;
