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Y is the given axiom, and Y → Z is the other given axiom. Then applying modus ponens (MP), Z can be obtained. Now, we have two axioms, X, and Z with us. Premise 1 says that for all the statements form A and B, A → (B → A) id valid. Hypothetical syllogism can be understood like: if p, the q. If q, then r. So, if p, then r.
So, assuming X as A, and Z as B, putting in hypothetical syllogism to obtain the result:
X → (Z → X)
(Z → X) → Z, then
X → Z is a tautology or valid