Proof of Theorem CILTI

That the composition is a linear transformation was established in Theorem CLTLT, so we need only establish that the composition is injective. Applying Definition ILT, choose $\vect{x}$, $\vect{y}$ from $U$. Then if $\lt{\left(\compose{S}{T}\right)}{\vect{x}}=\lt{\left(\compose{S}{T}\right)}{\vect{y}}$, \begin{align*} &\Rightarrow&\lt{S}{\lt{T}{\vect{x}}}&=\lt{S}{\lt{T}{\vect{y}}} &&\knowl{./knowls/definition.LTC.knowl}{\text{Definition LTC}}\\ &\Rightarrow&\lt{T}{\vect{x}}&=\lt{T}{\vect{y}} &&\knowl{./knowls/definition.ILT.knowl}{\text{Definition ILT}}\text{ for }S\\ &\Rightarrow&\vect{x}&=\vect{y} &&\knowl{./knowls/definition.ILT.knowl}{\text{Definition ILT}}\text{ for }T \end{align*}