
theorem LC5aa:
  for n be Nat, a be Real st 1 < a holds
    a to_power n < a to_power (n+1)
  proof
    let n be Nat, a be Real;
    assume AS: 1 < a;
    LP3:a to_power (n+1) = (a to_power n)*(a to_power 1) by FIB_NUM2:5
    .=(a to_power n)*a by POWER:25;
    a to_power n > 0 by POWER:34,AS;then
    1*(a to_power n) < (a to_power n)*a by XREAL_1:68,AS;
    hence thesis by LP3;
  end;
