Proof of Theorem IILT

Because $T$ is invertible, Definition IVLT tells us there is a function $\ltdefn{\ltinverse{T}}{V}{U}$ such that \begin{align*} \compose{\ltinverse{T}}{T}&=I_U & \compose{T}{\ltinverse{T}}&=I_V \end{align*}

Additionally, Theorem ILTLT tells us that $\ltinverse{T}$ is more than just a function, it is a linear transformation. Now view these two statements as properties of the linear transformation $\ltinverse{T}$. In light of Definition IVLT, they together say that $\ltinverse{T}$ is invertible (let $T$ play the role of $S$ in the statement of the definition). Furthermore, the inverse of $\ltinverse{T}$ is then $T$, i.e. $\ltinverse{\left(\ltinverse{T}\right)}=T$.