Mordell–Lang conjecture for Shimura varieties: Zariski-closure formulation

Let $S$ be a Shimura variety, let $obreak\boldsymbol{\Sigma} obreak\to S$ be a structure of finite rank, and call a weakly special subvariety $obreak\boldsymbol{\Sigma} obreak$-special when it contains a point of $obreak\boldsymbol{\Sigma} obreak$. Mordell–Lang conjecture for $S$. The Zariski closure of any \subset of $obreak\boldsymbol{\Sigma} obreak$ is a finite union of $obreak\boldsymbol{\Sigma} obreak$-special subvarieties. Equivalently, any subvariety of $S$ contains only finitely many maximal $obreak\boldsymbol{\Sigma} obreak$-special subvarieties. The paper proves that this formulation is equivalent to the preceding formulation of Mordell–Lang for Shimura varieties; neither is resolved in the supplied text.