
theorem
  for V being RealUnitarySpace, x,y being VECTOR of V holds ||.x+y.||^2
  + ||.x-y.||^2 = 2*||.x.||^2 + 2*||.y.||^2
proof
  let V be RealUnitarySpace;
  let x,y being VECTOR of V;
  ||.x+y.||^2 + ||.x-y.||^2 = ||.x.||^2 + 2 * x .|. y + ||.y.||^2 + ||.x-y
  .||^2 by Th29
    .= ||.x.||^2 + 2 * x .|. y + ||.y.||^2 + (||.x.||^2 - 2 * x .|. y + ||.y
  .||^2) by Th29
    .= ||.x.||^2 + ||.x.||^2 + 2*||.y.||^2;
  hence thesis;
end;
