Bezout implies gcd

Property-theoretic statement
The property of integral domains of being a Bezout domain, is stronger than the property of being a gcd domain.

Proof outline

 * Pick two elements in the given Bezout domain. We need to determine their gcd.
 * Consider the ideal generated by these two elements. This is a finitely generated ideal, and by the definition of Bezout domain, is a principal ideal. Pick a generator for this principal ideal.
 * Show that this generator is indeed the gcd of the two elements.