
theorem Th1:
  for A being set, f being FinSequence of bool A st
  (for i being Nat st 1 <= i & i < len f holds f/.i c= f/.(i+1))
 for i, j being Nat st i <= j & 1 <= i & j <= len f holds f/.i c= f/.j
proof
  let A be set;
  let f be FinSequence of bool A such that
A1: for i being Nat st 1 <= i & i < len f holds f/.i c= f/.(i +1);
  let i, j be Nat such that
A2: i <= j and
A3: 1 <= i and
A4: j <= len f;
  consider k be Nat such that
A5: i+k = j by A2,NAT_1:10;
  defpred P[Nat] means i+$1 <= j implies f/.i c= f/.(i+$1);
A6: now
    let k be Nat;
A7: i+k+1 = i+(k+1);
    assume
A8: P[k];
    thus P[k+1]
    proof
      i+k >= i by NAT_1:11;
      then
A9:   i+k >= 1 by A3,XXREAL_0:2;
      assume
A10:  i+(k+1) <= j;
      then i+k < j by A7,NAT_1:13;
      then i+k < len f by A4,XXREAL_0:2;
      then f/.(i+k) c= f/.(i+(k+1)) by A1,A7,A9;
      hence thesis by A7,A8,A10,NAT_1:13;
    end;
  end;
A11: P[0];
A12: for k being Nat holds P[k] from NAT_1:sch 2(A11,A6);
  thus thesis by A12,A5;
end;
