
theorem Th37:
  for R being non empty RelStr, s being sequence of R, j being Nat
  st s is descending holds s^\j is descending
proof
  let R be non empty RelStr, s1 be sequence of R, j be Nat
  such that
A1: s1 is descending;
  set s2 = s1^\j;
  set IR = the InternalRel of R;
  now
    let n be Nat;
    set nj = n+j;
A2: s2.n = s1.nj by NAT_1:def 3;
A3: s2.(n+1) = s1.((n+1)+j) by NAT_1:def 3
      .= s1.(n+j+1);
    hence s2.(n+1) <> s2.n by A1,A2;
    thus [s2.(n+1), s2.n] in IR by A1,A2,A3;
  end;
  hence thesis;
end;
