Theorem 1 (« Bezout's identity »)

— Elements $x_i \ (i \in I)$ of a principal ideal domain $A$ are setwise coprime if and only if there exist elements $a_i \ (i \in I)$ of $A$, all but finitely many of them zero, such that $\sum a_i x_i = 1$.

The necessity of the condition is just Prop. 1. Conversely, if $\sum a_i x_i = 1$ then every common divisor of the $x_i$ divides 1, and so 1 is a gcd of the $x_i$.