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.| > 0
proof
  p <> 0.TOP-REAL n implies |.p.| > 0
  proof
    assume
A1: p <> 0.TOP-REAL n;
    n in NAT & 0 <= |.p.| by Th36,ORDINAL1:def 12;
    hence thesis by A1,TOPRNS_1:24,XXREAL_0:1;
  end;
  hence thesis by Th37;
end;
