
theorem
  for k,n be Nat st k <= n
    holds Big_Oh( seq_n^k ) c= Big_Oh( seq_n^(n) )
  proof
    let k,n be Nat;
    assume k <= n;then
    consider i be Nat such that LA: n = k + i by NAT_1:10;
    defpred P[Nat] means
    Big_Oh( seq_n^k ) c= Big_Oh( seq_n^(k+ $1) );
    P0:P[0];
    P1:for x be Nat st P[x] holds P[x+1]
    proof
      let x be Nat;
      assume P1L1:P[x];
      Big_Oh( seq_n^(k+x) ) c= Big_Oh( seq_n^(k+ x +1) ) by LMXFIN9;
      hence thesis by XBOOLE_1:1,P1L1;
    end;
    for x be Nat holds P[x] from NAT_1:sch 2(P0,P1);
    hence thesis by LA;
  end;
