
theorem LM090:
  for n be Nat,
  x be Element of NAT,
  y be Tuple of n, BOOLEAN
  st y = Nat2BL.x
  holds
  n = LenBSeq x & Absval y = x & Nat2BL. Absval y = y
  proof
    let n be Nat,
    x be Element of NAT,
    y be Tuple of n, BOOLEAN;
    assume AS: y = Nat2BL.x;
    A1:Nat2BL.x = (LenBSeq x) -BinarySequence x by Def2;
    A3: x < 2 to_power (LenBSeq x) by LM006;
    len y = (LenBSeq x) by CARD_1:def 7,AS,A1;
    hence n = (LenBSeq x) by CARD_1:def 7;
    hence thesis by A3,BINARI_3:35,A1,AS;
  end;
