reserve n,m,k for Nat,
  x,y for set,
  r for Real;
reserve C,D for non empty finite set,
  a for FinSequence of bool D;

theorem Th8:
  for a be terms've_same_card_as_number FinSequence of bool D st 1
  <=n & n<=len a - 1 holds a.(n+1) \ a.n <> {}
proof
  let A be terms've_same_card_as_number FinSequence of bool D;
  assume that
A1: 1<=n and
A2: n<=len A - 1;
A3: n+1<=len A by A2,XREAL_1:19;
  n<=n+1 by NAT_1:11;
  then
A4: n<=len A by A3,XXREAL_0:2;
  then reconsider An1 = A.(n+1), An = A.n as finite set by A1,A3,Lm2,NAT_1:11;
  1<=n+1 by NAT_1:11;
  then
A5: card (An1) = n+1 by A3,Def1;
  assume A.(n+1) \ A.n = {};
  then
A6: A.(n+1) c= A.n by XBOOLE_1:37;
  card (An) = n by A1,A4,Def1;
  then n+1 <= n by A5,A6,NAT_1:43;
  then 1<=n-n by XREAL_1:19;
  then 1<=0;
  hence contradiction;
end;
