Proof of Theorem RPI

(⇒) If $\vect{v}\in\rng{T}$, then there is a vector $\vect{u}\in U$ such that $\lt{T}{\vect{u}}=\vect{v}$. This qualifies $\vect{u}$ for membership in $\preimage{T}{\vect{v}}$, and thus the preimage of $\vect{v}$ is not empty.

(⇐) Suppose the preimage of $\vect{v}$ is not empty, so we can choose a vector $\vect{u}\in U$ such that $\lt{T}{\vect{u}}=\vect{v}$. Then $\vect{v}\in\rng{T}$.