We are asked to prove that $\krn{T}$ is a subset of $\krn{\compose{S}{T}}$. Employing Definition SSET, choose $\vect{x}\in\krn{T}$. Then we know that $\lt{T}{\vect{x}}=\zerovector$. So
\begin{align*}
\lt{\left(\compose{S}{T}\right)}{\vect{x}}&=\lt{S}{\lt{T}{\vect{x}}}&&\knowl{./knowls/definition.LTC.knowl}{\text{Definition LTC}}\\
&=\lt{S}{\zerovector}&&\vect{x}\in\krn{T}\\
&=\zerovector&&\knowl{./knowls/theorem.LTTZZ.knowl}{\text{Theorem LTTZZ}}
\end{align*}
This qualifies $\vect{x}$ for membership in $\krn{\compose{S}{T}}$.