
theorem Th25:
  for b,i be Nat st b >= 1 holds (powerfact b).i <= ((1/b) GeoSeq).i
  proof
    let b,i be Nat;
    assume
A1: b >= 1;
A3: (powerfact b).i = 1/(b to_power (i!)) by DefPower;
A2: ((1/b) GeoSeq).i = (1/b)|^i by PREPOWER:def 1
                    .= 1/(b to_power i) by PREPOWER:7;
    1 * (b to_power i) <= 1 * (b to_power (i!)) by A1,PRE_FF:8,NEWTON:38;
    hence thesis by A2,A3,A1,XREAL_1:102;
  end;
