Prove a set is well-ordered

Let $FS(α, β) ⇋ { f | f : β->α & Fin( {γ | γ ∈ β & not( f(γ) = 0 ) })}$ for all ordinals α, β,
where Fin(A) means the set A is finite.

We define a relation ≺ on FS:  $f ≺ g ←→ (∃γ < β)( f(γ) < g(γ) & (∀δ < β)( γ < δ ⇒ f(δ) = g(δ)))$.

How to prove that (FS, ⪯) is well-ordered?

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