Common Divisor in Integral Domain Divides Linear Combination

Theorem
Let $\struct {D, +, \times}$ be an integral domain.

Let $c$ be a common divisor of two elements $a$ and $b$ of $D$.

That is:
 * $a, b, c \in D: c \divides a \land c \divides b$

Then:


 * $\forall p, q \in D: c \divides \paren {p \times a + q \times b}$