
theorem
  for R being non empty antisymmetric transitive RelStr,
  f being sequence of R st f is descending
  holds for j, i being Nat st i<j
  holds f.i<>f.j & [f.j,f.i] in the InternalRel of R
proof
  let R be non empty antisymmetric transitive RelStr,
  f be sequence of R such that
A1: f is descending;
  set IR = the InternalRel of R, CR = the carrier of R;
A2: IR is_transitive_in CR by ORDERS_2:def 3;
A3: IR is_antisymmetric_in CR by ORDERS_2:def 4;
  defpred P[Nat] means (for i being Nat st i < $1
  holds f.i <> f.$1 & [f.$1, f.i] in IR);
A4: P[ 0 ];
  now
    let j be Nat such that
A5: for i being Nat st i < j holds f.i <> f.j & [f.j, f.i] in IR;
    let i be Nat such that
A6: i < j+1;
    now per cases by XXREAL_0:1;
      suppose i > j;
        hence f.i <> f.(j+1) & [f.(j+1), f.i] in IR by A6,NAT_1:13;
      end;
      suppose i = j;
        hence f.i <> f.(j+1) & [f.(j+1), f.i] in IR by A1;
      end;
      suppose i < j;
        then
A7:     [f.j, f.i] in IR by A5;
A8:     f.(j+1)<>f.j by A1;
        [f.(j+1), f.j] in IR by A1;
        hence f.i <> f.(j+1) & [f.(j+1), f.i] in IR by A2,A3,A7,A8;
      end;
    end;
    hence f.i <> f.(j+1) & [f.(j+1), f.i] in IR;
  end;
  then
A9: for j being Nat st P[j] holds P[j+1];
  thus for j being Nat holds P[j] from NAT_1:sch 2(A4,A9);
end;
