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 Th38:
  |(p,p)| = 0 iff |.p.| = 0
proof
A1: |(p,p)| = 0^2 implies |.p.| = 0
  proof
    assume |(p,p)| = 0^2;
    then |.p.|^2 = 0 by Th34;
    hence thesis by XCMPLX_1:6;
  end;
  |.p.| = 0 implies |(p,p)| = 0^2 by Th34;
  hence thesis by A1;
end;
