Lattcies of invariant subspaces for a quasiaffine transform of a unilateral shift of finite multiplicity

Let $T$ be a contraction, let $S$ be an unilateral shift of finite multiplicity, and let $X$ be an operator with zero kernel, dense range, and such that $XT=SX$. Then the mapping $E\mapsto\text{clos}XE$, $E\in\text{Lat}T$, is an isomorphism between the latticies $\text{Lat}T$ and $\text{Lat}S$ of invariant subspaces of $T$ and $S$.