
theorem Th3:
  for k be Nat st 2 <= k holds 4 <= Radix(k)
proof
  defpred P[Nat] means 4 <= Radix($1);
A1: for kk be Nat st kk >= 2 & P[kk] holds P[kk + 1]
  proof
    let kk be Nat;
    assume that
    2 <= kk and
A2: 4 <= Radix(kk);
    Radix(kk + 1) = 2 to_power (kk + 1) by RADIX_1:def 1
      .= 2 to_power (1) * 2 to_power (kk) by POWER:27
      .= 2 * 2 to_power (kk) by POWER:25
      .= 2 * Radix(kk) by RADIX_1:def 1;
    then Radix(kk + 1) >= Radix(kk) by XREAL_1:151;
    hence thesis by A2,XXREAL_0:2;
  end;
  Radix(2) = 2 to_power (1+1) by RADIX_1:def 1
    .= 2 to_power 1 * 2 to_power 1 by POWER:27
    .= 2 * (2 to_power 1) by POWER:25
    .= 2 * 2 by POWER:25;
  then
A3: P[2];
  for i being Nat st 2 <= i holds P[i] from NAT_1:sch 8(A3,A1);
  hence thesis;
end;
