State and prove Hartogs' lemma. [You may assume the result that any well-ordered set is isomorphic to a unique ordinal.]

Let $a$ and $b$ be sets such that there is a bijection $a \sqcup b \rightarrow a \times b$. Show, without assuming the Axiom of Choice, that there is either a surjection $b \rightarrow a$ or an injection $b \rightarrow a$. By setting $b=\gamma(a)$ (the Hartogs ordinal of $a$ ) and considering $(a \sqcup b) \times(a \sqcup b)$, show that the assertion 'For all infinite cardinals $m$, we have $m^{2}=m^{\prime}$ implies the Axiom of Choice. [You may assume the Cantor-Bernstein theorem.]