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
  for x,y being real-valued FinSequence st len x=len y holds |.x+y.|^2 + |.x
  -y.|^2 = 2*(|.x.|^2 + |.y.|^2)
proof
  let x,y be real-valued FinSequence;
  assume
A1: len x=len y;
  then
  |.x+y.|^2 + |.x-y.|^2 = (|.x.|^2 + 2*|(y, x)| + |.y.|^2) + |.x-y.|^2 by Th10
    .= (|.x.|^2 + 2*|(x, y)| + |.y.|^2) + (|.x.|^2 - 2*|(y, x)| + |.y.|^2)
  by A1,Th11
    .= 2*(|.x.|^2 + |.y.|^2);
  hence thesis;
end;
