reserve c, c1, c2, d, d1, d2, e, y for Real,
  k, n, m, N, n1, N0, N1, N2, N3, M for Element of NAT,
  x for set;

theorem
  ex F being FUNCTION_DOMAIN of NAT,REAL st F = { seq_n^(1) } & (for n
  holds (seq_n^(-1)).n <= (seq_n^(1)).n) & not seq_n^(-1) in F to_power Big_Oh(
  seq_const(1))
proof
  set t = seq_n^(-1);
  reconsider F = { seq_n^(1) } as FUNCTION_DOMAIN of NAT,REAL by FUNCT_2:121;
  take F;
  thus F = { seq_n^(1) };
A1: now
    let n;
    per cases;
    suppose
A2:   n = 0;
      then (seq_n^(-1)).n = 0 by Def3;
      hence (seq_n^(-1)).n <= (seq_n^(1)).n by A2,Def3;
    end;
    suppose
A3:   n > 0;
      then
A4:   n >= 0+1 by INT_1:7;
A5:   n to_power (-1) <= n to_power 1
      proof
        per cases by A4,XXREAL_0:1;
        suppose
A6:       n = 1;
          then n to_power (-1) = 1 by POWER:26;
          hence thesis by A6,POWER:26;
        end;
        suppose
          n > 1;
          hence thesis by PRE_FF:8;
        end;
      end;
      (seq_n^(-1)).n=n to_power (-1) by A3,Def3;
      hence (seq_n^(-1)).n <= (seq_n^(1)).n by A3,A5,Def3;
    end;
  end;
  now
    assume
A7: t in F to_power Big_Oh(seq_const(1));
    ex H being FUNCTION_DOMAIN of NAT,REAL st H = F &( t in H to_power
Big_Oh(seq_const(1)) iff ex N,c,k st c>0 & for n st n >= N holds 1 <= t.n & t.n
    <= c*(seq_n^(k)).n) by Th9;
    then consider N0,c,k such that
    c>0 and
A8: for n st n >= N0 holds 1 <= t.n & t.n <= c*(seq_n^(k)).n by A7;
    set N = max(N0,2);
A9: N >= 2 by XXREAL_0:25;
A10: N >= N0 by XXREAL_0:25;
    now
      let n;
      assume
A11:  n >= N;
      then n >= 2 by A9,XXREAL_0:2;
      then
A12:  n > 1 by XXREAL_0:2;
      n >= N0 by A10,A11,XXREAL_0:2;
      then
A13:  t.n >= 1 by A8;
      t.n = n to_power (-1) by A9,A11,Def3;
      hence contradiction by A13,A12,POWER:36;
    end;
    hence contradiction;
  end;
  hence thesis by A1;
end;
