reserve n for non zero Nat,
  j,k,l,m for Nat,
  g,h,i for Integer;

theorem Th2:
  for m being Nat holds 2 to_power m >= m
proof
  defpred P[Nat] means 2 to_power $1 >= $1;
A1: for m being Nat st P[m] holds P[m+1]
  proof
    let m be Nat such that
A2: 2 to_power m >= m;
    per cases;
    suppose
A3:   m = 0;
      then 2 to_power (m+1) = 2 by POWER:25;
      hence thesis by A3;
    end;
    suppose
A4:   m > 0;
      reconsider m2 = 2 to_power m as Nat;
      m2 * 2 >= m * 2 & 2 to_power 1 = 2 by A2,NAT_1:4,POWER:25;
      then
A5:   2 to_power (m+1) >= m * 2 by POWER:27;
      m * 2 >= m + 1 by A4,Th1;
      hence thesis by A5,XXREAL_0:2;
    end;
  end;
A6: P[0];
  thus for m being Nat holds P[m] from NAT_1:sch 2(A6,A1);
end;
