
theorem LM0071:
  for n be Nat,
  x be Tuple of n+1, BOOLEAN st x.(n+1) = 1
  holds
  2 to_power n <= Absval(x) & Absval(x) < 2 to_power (n + 1)
  proof
    let n be Nat,
    x be Tuple of n+1, BOOLEAN;
    assume AS: x.(n+1) = 1;
    len x = n+1 by CARD_1:def 7; then
    n+1 in Seg len x by FINSEQ_1:4; then
    n+1 in dom x by FINSEQ_1:def 3; then
    P3: x/.(n+1) = 1 by AS,PARTFUN1:def 6;
    0 <= (n+1) -1; then
    (n+1) -' 1 = n by XREAL_0:def 2;
    hence 2 to_power n <= Absval(x) by BINARI_4:12,P3;
    thus Absval(x) < 2 to_power (n + 1) by BINARI_3:1;
  end;
