
theorem Th18:
  for f being Function st (for i being Nat holds f.(i+1
  ) c= f.i) for i, j being Nat st i <= j holds f.j c= f.i
proof
  let f be Function;
  assume
A1: for i being Nat holds f.(i+1) c= f.i;
  let i, j be Nat;
  defpred P[Nat] means i + $1 <= j implies f.(i + $1) c= f.i;
A2: now
    let k be Nat;
    assume
A3: P[k];
A4: i + k + 1 = i + (k + 1);
    then f.(i + (k + 1)) c= f.(i + k) by A1;
    hence P[k+1] by A4,A3,NAT_1:13,XBOOLE_1:1;
  end;
A5: P[0];
A6: for k being Nat holds P[k] from NAT_1:sch 2(A5,A2);
  assume i <= j;
  then consider k be Nat such that
A7: i + k = j by NAT_1:10;
  thus thesis by A6,A7;
end;
