
theorem Th8:
  for R being non empty transitive RelStr, f being sequence of R
  st f is non-increasing holds for j, i being Nat st i<j
  holds [f.j,f.i] in the InternalRel of R
proof
  let R be non empty transitive RelStr, f be sequence of R such that
A1: f is non-increasing;
  set IR = the InternalRel of R, CR = the carrier of R;
A2: IR is_transitive_in CR by ORDERS_2:def 3;
  defpred P[Nat] means (for i being Nat st i < $1
  holds [f.$1, f.i] in IR);
A3: P[ 0 ];
  now
    let j be Nat such that
A4: for i being Nat st i < j holds [f.j, f.i] in IR;
    let i be Nat such that
A5: i < j+1;
    now per cases by XXREAL_0:1;
      suppose i > j;
        hence [f.(j+1), f.i] in IR by A5,NAT_1:13;
      end;
      suppose i = j;
        hence [f.(j+1), f.i] in IR by A1;
      end;
      suppose i < j;
        then
A6:     [f.j, f.i] in IR by A4;
        [f.(j+1), f.j] in IR by A1;
        hence [f.(j+1), f.i] in IR by A2,A6;
      end;
    end;
    hence [f.(j+1), f.i] in IR;
  end;
  then
A7: for j being Nat st P[j] holds P[j+1];
  thus for j being Nat holds P[j] from NAT_1:sch 2(A3,A7);
end;
