How to prove that a version of adapted process X is also adapted wrt Filtration?

Question

We have a process $X_1$ and a modification of that process $X_2$.

Criteria to be a modification is as follows: P($X_1t$ = $X_2t$) = 1 for all t in interval I.

A filtration is just an increasing set of sigma-algebras.

How do I prove that if $X_1$ is adapted to filtration F, under the usual conditions, then $X_2$ as well?