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
  for a be ascending terms've_same_card_as_number FinSequence of bool D
  holds for n holds 1 <= n & n <= len a - 1 implies a.n <> a.(n+1)
proof
  let A be ascending terms've_same_card_as_number FinSequence of bool D;
  let n;
  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 n<=len A by A3,XXREAL_0:2;
  then
A4: n in dom A by A1,FINSEQ_3:25;
  1<=n+1 by NAT_1:11;
  then
A5: n+1 in dom A by A3,FINSEQ_3:25;
  n <> n+1;
  hence thesis by A4,A5,Th5;
end;
