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.|^2 + |.p-q.|^2 = 2*(|.p.|^2 + |.q.|^2)
proof
A1: (|.p.|^2 - 2*|(p, q)| + |.q.|^2) = (|.p.|^2 + |.q.|^2 -2*|(p, q)| );
  |.p+q.|^2 + |.p-q.|^2 = (|.p.|^2 + 2*|(p, q)| + |.q.|^2) + |.p-q.|^2 by Th43
    .= (|.p.|^2 + |.q.|^2 + 2*|(p, q)|) + (|.p.|^2 + |.q.|^2 - 2*|(p, q)|)
  by A1,Th44
    .= 2*(|.p.|^2 + |.q.|^2);
  hence thesis;
end;
