How do I translate this sentence from EGA I, part (3.6.2)?

I need some help translating the following sentence to English. It is from (3.6.2) in EGA I.

Si $U,V$ sont deux ouverts non vides dans $X$, $U \cap V$ est non vide, donc $\mathcal{F}(X) \to \mathcal{F}(U) \to \mathcal{F}(U \cap V)$ et $\mathcal{F}(X) \to \mathcal{F}(U)$ étant des isomorphismes, il en est de même de $\mathcal{F}(U) \to \mathcal{F}(U \cap V)$ et de même $\mathcal{F}(V) \to \mathcal{F}(U \cap V)$ est un isomorphisme.

My attempt:

If $U,V$ are two non-empty open subsets of $X$, $U \cap V$ is non-empty, so $\mathcal{F}(X) \to \mathcal{F}(U) \to \mathcal{F}(U \cap V)$ and $\mathcal{F}(X) \to \mathcal{F}(U)$ being isomorphisms, it is the same with $\mathcal{F}(U) \to \mathcal{F}(U \cap V)$ and similarly $\mathcal{F}(V) \to \mathcal{F}(U \cap V)$ is an isomorphism.

In particular, I'm not sure if the part about $U \cap V$ should be "and if $U \cap V$ is non-empty" or whether I've mistranslated the repeated "de même" construction.

Your translation is good. $U\cap V$ is not empty here because $X$ is irreducible. This proposition starts with suppose that $X$ is irreducible...