
theorem Th3:
  for R being non empty transitive RelStr, f being sequence of R
  st f is weakly-ascending
  for i,j being Nat st i < j holds f.i <= f.j
proof
  let R be non empty transitive RelStr, f be sequence of R such that
A1: f is weakly-ascending;
  let i be Nat;
  defpred P[Nat] means i < $1 implies f.i <= f.$1;
A2: P[ 0 ] by NAT_1:2;
A3: for j being Nat st P[j] holds P[j+1]
  proof
    let j be Nat such that
A4: P[j] and
A5: i < j+1;
    reconsider fj1 = f.(j+1) as Element of R;
A6: [f.j, f.(j+1)] in the InternalRel of R by A1;
    then
A7: f.j <= fj1;
A8: i <= j by A5,NAT_1:13;
    per cases by A8,XXREAL_0:1;
    suppose i < j;
      hence thesis by A4,A7,ORDERS_2:3;
    end;
    suppose i = j;
      hence thesis by A6;
    end;
  end;
  thus for j being Nat holds P[j] from NAT_1:sch 2(A2,A3);
end;
