reserve k, m, n, p, K, N for Nat;
reserve i for Integer;
reserve x, y, eps for Real;
reserve seq, seq1, seq2 for Real_Sequence;
reserve sq for FinSequence of REAL;

theorem Th38:
  x=1/(n+1) implies (n!)/((n+k+1)!)<=x ^ (k+1)
proof
  defpred P[Nat] means (n!)/((n+$1+1)!)<=x ^ ($1+1);
  assume
A1: x=1/(n+1);
A2: now
    let k;
    assume
A3: P[k];
A4: (n!)/((n+(k+1)+1)!) = (n!)*((n+(k+1)+1)!)"
      .= (n!)*((n+(k+1)+1)*((n+k+1)!))" by NEWTON:15
      .= (n!)*((n+(k+1)+1)"*((n+k+1)!)") by XCMPLX_1:204
      .= (n!)*((n+k+1)!)"*(n+(k+1)+1)"
      .= (n!)/((n+k+1)!)*(n+(k+1)+1)";
    n+(k+1)>=n+0 by XREAL_1:6;
    then n+(k+1)+1>=n+1 by XREAL_1:6;
    then
A5: (n+(k+1)+1)"<=1/(n+1) by XREAL_1:85;
    x ^ (k+1)*x = x ^ (k+1)*(x^1)
      .= x ^ ((k+1)+1) by A1,POWER:27;
    hence P[k+1] by A1,A3,A5,A4,XREAL_1:66;
  end;
  (n!)/((n+1)!) = (n!)/((n+1)*(n!)) by NEWTON:15
    .= (n!)*((n+1)*(n!))"
    .= (n!)*((n+1)"*(n!)") by XCMPLX_1:204
    .= (n!)*(n!)"*(n+1)"
    .= 1*(n+1)" by XCMPLX_0:def 7
    .= x by A1;
  then
A6: P[0];
  for n holds P[n] from NAT_1:sch 2(A6,A2);
  hence thesis;
end;
