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 <> 0.TOP-REAL n iff |(p, p)| > 0
proof
  p <> 0.TOP-REAL n implies |(p, p)| > 0
  proof
    assume p <> 0.TOP-REAL n;
    then
A1: |(p,p)| <> 0 by Th39;
    0 <= |(p,p)| by Th33;
    hence thesis by A1,XXREAL_0:1;
  end;
  hence thesis by Th39;
end;
