Problem in the CLRS Linear Programming chapter

I’m currently reading the CLRS Linear Programming chapter and there is something i don’t understand.

The goal is to prove that given a basic set of variables, the associated slack form is unique

They first prove a lemma :

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And then they prove the result :

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My concern is that to prove the second lemma, they apply the first lemma. However equations (29.79) -> (29.82) only holds for feasable solutions, which is not for any x, so why can they apply the first lemma ?