
theorem LMExtBit500:
  for K be non zero Nat,
  x,y be Tuple of K, BOOLEAN
  holds
  x + y = y + x
  proof
    let K be non zero Nat,
    x,y be Tuple of K, BOOLEAN;
    for i being Nat st i in Seg K holds
    (x+y) /. i = ((y /. i) 'xor' (x /. i)) 'xor' ((carry (y,x)) /. i)
    proof
      let i be Nat;
      assume i in Seg K; then
      (x+y) /. i = ((x /. i) 'xor' (y /. i)) 'xor' ((carry (x,y)) /. i)
      by BINARITH:def 5;
      hence
      (x+y) /. i = ((y /. i) 'xor' (x /. i)) 'xor' ((carry (y,x)) /. i)
      by LMExtBit501;
    end;
    hence x+y =y+x by BINARITH:def 5;
  end;
