  The aim of this note is to provide a short and self-contained proof of
H\"ormander's theorem about the smoothness of transition probabilities for a
diffusion under H\"ormander's "brackets condition". While both the result and
the technique of proof are well-known, the exposition given here is novel in
two aspects. First, we introduce Malliavin calculus in an "intuitive" way,
without using Wiener's chaos decomposition. While this may make it difficult to
prove some of the standard results in Malliavin calculus (boundedness of the
derivative operator in $L^p$ spaces for example), we are able to bypass these
and to replace them by weaker results that are still sufficient for our
purpose. Second, we introduce a notion of "almost implication" and "almost
truth" (somewhat similar to what is done in fuzzy logic) which allows, once the
foundations of Malliavin calculus are laid out, to give a very short and
streamlined proof of H\"ormader's theorem that focuses on the main ideas
without clouding it by technical details.
