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 Th39:
  |(p,p)| = 0 iff p = 0.TOP-REAL n
proof
  |(p,p)| = 0 implies p = 0.TOP-REAL n
  proof
    assume |(p,p)| = 0;
    then n in NAT & |.p.| = 0 by Th38,ORDINAL1:def 12;
    hence thesis by TOPRNS_1:24;
  end;
  hence thesis by Th30;
end;
