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,q)| = (1/4)*(|.p+q.|^2 - |.p-q.|^2)
proof
  |.p+q.|^2 - |.p-q.|^2 = (|.p.|^2 + 2*|(p,q)| + |.q.|^2) - |.p-q.|^2 by Th43
    .= (|.p.|^2 + 2*|(p, q)| + |.q.|^2) - (|.p.|^2 - 2*|(p, q)| + |.q.|^2)
  by Th44
    .= 4*|(p, q)|;
  hence thesis;
end;
