End Double_round_sqrt_beta_ge_4_FTZ.
End Double_round_sqrt_beta_ge_4.
End Double_round_sqrt.
Section Double_round_div.

Lemma ln_beta_div_disj :
forall x y : R,
0 < x -> 0 < y ->
((ln_beta (x / y) = ln_beta x - ln_beta y :> Z)%Z
\/ (ln_beta (x / y) = ln_beta x - ln_beta y + 1 :> Z)%Z).