A nice Set Theory problem

Given sets A and B.

Consider the sets A' = A \times \{0,1\} and B' = B \times \{0,1\}. That is, A' is the set of all ordered pairs (a,\varepsilon) where a \in A and \varepsilon \in \{0,1\}. Similarly for B'.

(We deduce that the cardinality, or ‘size’, of A' is two times that of A.)

It is known that there exists a bijection from A' to B'. Prove that there exists a bijection from A to B.

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