The finitely generated-field reduction for full Mordell–Lang

Let $L_0$ be a field finitely generated over ${\mathbb F}_p$, let $L$ be an algebraic closure of $L_0$, and let $L_0^\text{per}$ be the perfect closure of $L_0$ in $L$. Let $A$ be a semiabelian variety over $L_0$, let $X\subset A$ be an irreducible subvariety, and let $\Gamma\subset A(L_0^\text{per})$ be a subgroup of finite rank. The finitely generated-field reduction. If $X(L_0^\text{per})\cap\Gamma$ is Zariski dense in $X_L$, then $X_L$ is a special subvariety of $A_L$. The source presents this statement as a consequence sufficient to imply the full Mordell–Lang conjecture, and proves equivalence with the other reduction for ordinary or supersingular abelian varieties.