  In this paper we continue the investigation started in [A.M.St.-Small],
dealing with bialgebras $A$ with an $H$-bilinear coalgebra projection over an
arbitrary subbialgebra $H$ with antipode. These bialgebras can be described as
deformed bosonizations $R#_{\xi} H$ of a pre-bialgebra $R$ by $H$ with a
cocycle $\xi$. Here we describe the behavior of $\xi$ in the case when $R$ is
f.d. and thin i.e. it is connected with one dimensional space of primitive
elements. This is used to analyze the arithmetic properties of $A$. Meaningful
results are obtained when $H$ is cosemisimple. By means of Ore extension
construction, we provide some examples of atypical situations (e.g. the
multiplication of $R$ is not $H$-colinear or $\xi$ is non-trivial).
