
theorem LMExtBit9:
  for K be non zero Nat,
  x,y be Tuple of K, BOOLEAN
  st x.(len x) =1 & y.(len y) =1
  holds not x, y are_summable
  proof
    let K be non zero Nat,
    x,y be Tuple of K, BOOLEAN;
    assume AS: x.(len x) =1 & y.(len y) =1;
    A5: len x = K & len y = K by CARD_1:def 7;
    1<=len x by NAT_1:25; then
    len x in dom x by FINSEQ_3:25; then
    A6: x /. K = 1 by AS,A5,PARTFUN1:def 6;
    1<=len y by NAT_1:25; then
    len y in dom y by FINSEQ_3:25; then
    A7: y /. K = 1 by AS,A5,PARTFUN1:def 6;
    (((x /. K) '&' (y /. K)) 'or' ((x /. K) '&' ((carry (x,y)) /. K)))
    'or' ((y /. K) '&' ((carry (x,y)) /. K)) = 1 by A6,A7; then
    add_ovfl (x,y) = 1 by BINARITH:def 6;
    hence thesis by BINARITH:def 7;
  end;
