
theorem LMExtBit61:
  for K be non zero Nat,
  x,y be Tuple of K, BOOLEAN
  st x, y are_summable
  holds y, x are_summable
  proof
    let K be non zero Nat,
    x,y be Tuple of K, BOOLEAN;
    assume x, y are_summable; then
    add_ovfl (x,y) = 0 by BINARITH:def 7; then
    P1: (((x /. K) '&' (y /. K)) 'or' ((x /. K) '&' ((carry (x,y)) /. K)))
    'or' ((y /. K) '&' ((carry (x,y)) /. K)) = 0 by BINARITH:def 6;
    carry (x,y) /.K =carry (y,x) /.K by LMExtBit501; then
    (((y /. K) '&' (x /. K)) 'or' ((y /. K) '&' ((carry (y,x)) /. K)))
    'or' ((x /. K) '&' ((carry (y,x)) /. K)) = 0 by P1; then
    add_ovfl (y,x) = 0 by BINARITH:def 6;
    hence thesis by BINARITH:def 7;
  end;
