Strengthening of de Finetti's coherence for conditional probability (or prevision) and extensions of the latter from a given class of events (or bounded random quantities) to any other larger class have been repeatedly given in the relevant literature (besides de Finetti himself for the strengthening in the case of conditioning with respect to any given event). In this article a clear, detailed and complete exposition of the two aforementioned important questions is given. The results are essentially known: we quote, e.g., the papers by \textit{R. S. Lehman}, J. Symb. Logic 20, 251-262 (1955; Zbl 0066.110) and \textit{P. M. Williams}, Notes on conditional previsions. Tech. Rep., Univ. Sussex (1975), which are not cited in the article.