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

theorem
  for n be non zero Nat holds 0 <= 2 to_power (n-'1) - 1 & -(2 to_power
  (n-'1)) <= 0
proof
  defpred P[Nat] means 0 <= 2 to_power ($1-'1) - 1 & -(2 to_power ($1-'1)) <=
  0;
A1: for k be non zero Nat st P[k] holds P[k+1]
  proof
    let k be non zero Nat such that
A2: 0 <= 2 to_power (k-'1) - 1 and
    -(2 to_power (k-'1)) <= 0;
    (k+1)-'1 = k by NAT_D:34;
    then 2 to_power (k-'1) < 2 to_power ((k+1)-'1) by NAT_2:9,POWER:39;
    hence thesis by A2,XREAL_1:9;
  end;
  1 - 1 = 0;
  then 2 to_power (1-'1) = 2 to_power 0 by XREAL_0:def 2
    .= 1 by POWER:24;
  then
A3: P[1];
  thus for n being non zero Nat holds P[n] from NAT_1:sch 10(A3,A1);
end;
