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 Th44:
  |.p-q.|^2 = |.p.|^2 - 2*|(q, p)| + |.q.|^2
proof
  |.p-q.|^2 = |(p-q, p-q)| by Th34
    .= |(p, p)| - 2*|(q, p)| + |(q, q)| by Th29
    .= |.p.|^2 - 2*|(q, p)| + |(q, q)| by Th34
    .= |.p.|^2 - 2*|(q, p)| + |.q.|^2 by Th34;
  hence thesis;
end;
