Common Divisor in Integral Domain Divides Linear Combination

Theorem
Let $\left({D, +, \times}\right)$ be an integral domain.

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

That is:
 * $a, b, c \in D: c \mathop \backslash a \land c \mathop \backslash b$.

Then:


 * $\forall p, q \in D: c \mathop \backslash \left({p \times a + q \times b}\right)$

Corollary
Let $c$ be a common divisor of two integers $a$ and $b$.

That is:
 * $a, b, c \in \Z: c \mathop \backslash a \land c \mathop \backslash b$.

Then $c$ divides any integer combination of $a$ and $b$:


 * $\forall p, q \in \Z: c \mathop \backslash \left({p a + q b}\right)$

Proof of Corollary
Follows directly from Integers form Integral Domain.