[dwheeler]

This is a decent intuitive explanation.

If you want an absolutely rigorous proof, you can view this Metamath proof: http://us.metamath.org/mpeuni/mulge0.html ; this has more far more steps, but is totally rigorous. It particular, its only axioms are those of classical logic and ZFC set theory (not even numbers are presumed, the system first proves "numbers exist and have these properties").