
theorem LMExtBit1:
  for K be Nat,
  x be Element of BOOLEAN*
  st len x <= K
  holds ExtBit(x,K+1) =ExtBit(x,K)^<* 0 *>
  proof
    let K be Nat,
    x be Element of BOOLEAN*;
    assume AS: len x <= K;
    A1: K + 0 <= K + 1 by XREAL_1:7;
    then AA1: len x <= K+1 by AS,XXREAL_0:2;
    A2:ExtBit(x,K) = x ^ ( 0* (K -' len x) ) by AS,Def20;
    A3:ExtBit(x,K+1) = x ^ ( 0* ((K+1) -' len x) ) by A1,Def20,AS,XXREAL_0:2;
    B1: (K -' len x) = K -len x by XREAL_0:def 2,AS,XREAL_1:48;
    ((K+1) -' len x) = (K+1) -len x by XREAL_0:def 2,AA1,XREAL_1:48; then
    (K+1) -' len x = (K -' len x)+1 by B1; then
    0* ((K+1) -' len x ) = (0* (K -' len x)) ^ <* 0 *> by FINSEQ_2:60;
    hence ExtBit(x,K+1)= ExtBit(x,K)^<* 0 *> by A2, FINSEQ_1:32,A3;
  end;
