Smallest Positive Integer Combination is Greatest Common Divisor

Theorem
Let $a, b \in \Z_{>0}$ be (strictly) positive integers.

Let $d \in \Z_{>0}$ be the smallest positive integer such that:
 * $d = a s + b t$

where $s, t \in \Z$.

Then:
 * $(1): \quad d \divides a \land d \divides b$
 * $(2): \quad c \divides a \land c \divides b \implies c \divides d$

where $\divides$ denotes divisibility.

That is, $d$ is the greatest common divisor of $a$ and $b$.