
theorem LMExtBit2:
  for K be non zero Nat,
  x,y be Tuple of K, BOOLEAN,
  x1,y1 be Tuple of K+1, BOOLEAN
  st x1=x^ <* 0 *>
  & y1=y^ <* 0 *>
  holds x1,y1 are_summable
  proof
    let K be non zero Nat,
    x,y be Tuple of K, BOOLEAN,
    x1,y1 be Tuple of K+1, BOOLEAN;
    set K1=K+1;
    assume A1:x1=x^ <* 0 *> & y1=y^ <* 0 *>;
    A5: len x = K & len y = K by CARD_1:def 7;
    A6: len x1 = K+1 & len y1 = K+1 by CARD_1:def 7;
    A7: K1 in Seg K1 by FINSEQ_1:4; then
    K1 in dom x1 by FINSEQ_1:def 3,A6; then
    x1/.K1 =x1.K1 by PARTFUN1:def 6; then
    A3:x1 /. K1 = 0 by FINSEQ_1:42,A1,A5;
    K1 in dom y1 by FINSEQ_1:def 3,A6,A7;
    then
    y1/.K1 =y1.K1 by PARTFUN1:def 6;
    then
    (((x1 /. K1) '&' (y1 /. K1)) 'or' ((x1 /. K1) '&' ((carry (x1,y1)) /. K1)))
    'or' ((y1 /. K1) '&' ((carry (x1,y1)) /. K1)) = 0 by A3,FINSEQ_1:42,A1,A5;
    then
    add_ovfl (x1,y1) = FALSE by BINARITH:def 6;
    hence thesis by BINARITH:def 7;
  end;
