
theorem
  for V being RealUnitarySpace, x,y being VECTOR of V st x,y
  are_orthogonal holds ||.x+y.||^2 = ||.x.||^2 + ||.y.||^2
proof
  let V be RealUnitarySpace;
  let x,y be VECTOR of V;
  assume x,y are_orthogonal;
  then
A1: x .|. y = 0 by BHSP_1:def 3;
  ||.x + y.||^2 = ||.x.||^2 + 2 * x .|. y + ||.y.||^2 by Th29;
  hence thesis by A1;
end;
