
theorem Th26:
  for n,k be Nat st k < 2 to_power n holds ((n+1)-BinarySequence k
  ).(n+1) = FALSE
proof
  let n,k be Nat;
A1: n+1-'1 = n+1-1 by NAT_D:37
    .= n;
  assume k < 2 to_power n;
  then
A2: (k div 2 to_power (n+1-'1)) mod 2 = 0 mod 2 by A1,NAT_D:27
    .= 0 by NAT_D:26;
A3: n + 1 in Seg (n+1) by FINSEQ_1:4;
  then n + 1 in Seg len ((n+1)-BinarySequence k) by CARD_1:def 7;
  then n + 1 in dom ((n+1)-BinarySequence k) by FINSEQ_1:def 3;
  hence ((n+1)-BinarySequence k).(n+1) = ((n+1)-BinarySequence k)/.(n+1) by
PARTFUN1:def 6
    .= IFEQ((k div 2 to_power (n+1-'1)) mod 2,0,FALSE,TRUE) by A3,Def1
    .= FALSE by A2,FUNCOP_1:def 8;
end;
