
theorem Th10:
  for d be Integer holds ex n be non zero Nat st 2 to_power (n-1) > d
  proof
    let d be Integer;
    per cases;
    suppose
A1:   d <= 0;
      take n = 1;
      thus thesis by A1,POWER:24;
    end;
    suppose
A2: d > 0; then
    d >= 1+0 by INT_1:7; then
    per cases by XXREAL_0:1;
    suppose
A3:   d = 1;
      take n = 2;
      2 to_power (n-1) = 2 to_power 1 .= 2;
      hence thesis by A3;
    end;
    suppose
      d > 1; then
      log (2,d) > 0 by ENTROPY1:4; then
A4:   [/ log (2,d) \] > 0 by INT_1:def 7;
      set n = [/ log (2,d) \] + 2;
      n in NAT by A4,INT_1:3; then
      reconsider n as Nat;
      reconsider n as non zero Nat by A4;
      take n;
      n-1 = [/ log (2,d) \] + 1; then
      2 to_power(n-1) > 2 to_power (log(2,d)) by POWER:39,INT_1:32;
      hence thesis by A2,POWER:def 3;
    end;
    end;
  end;
