How to prove Transitive property of Reducibility with a TM?
The properties that reductions have are:


They preserve language membership (or Yes-instances, or whichever phrasing you prefer).
They are computable.


So then given that you have two reductions $f$, which takes instances of $A$ and produces instances of $B$ such that $x \in A \Leftrightarrow f(x) \in B$, and $g$ which does the same for $B$ and $C$, you need to show:


$x \in A \Leftrightarrow g(f(x)) \in C$.
$f\cdot g$ is computable.


The first should be a fairly straightforward application of basic logic. The second is also easy.

You can't just take $A=HALT_{TM}$ though, as all you would show then is that $HALT_{TM} \leq_{m} C$, whereas your question (appears) to ask for a proof of the transitivity of many-one reductions in general. (As a comparison would showing that $(10^{100} &lt; b) \wedge (b &lt;c) \Rightarrow (10^{100} &lt; c)$ show that $(a &lt; b) \wedge (b &lt;c) \Rightarrow (a &lt; c)$?)