Linearly ordering the power set of a well ordered set with ZF (without AC)

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As the title says, my question is, how one can use only ZF-theory to prove that the power set of A, whereby (A, <) is a well-ordering, can be linearly ordered?

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Find the earliest element in the well order of A where they differ-where it is in one and not the other. Lexicographic order would take the one with the element first.