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 Th43:
  |.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 Th28
    .= |.p.|^2 + 2*|(q, p)| + |(q, q)| by Th34
    .= |.p.|^2 + 2*|(q, p)| + |.q.|^2 by Th34;
  hence thesis;
end;
