Theorem

The operation of multiplication on the set of integers $\Z$ is distributive over addition:

$\forall x, y, z \in \Z: x \times \paren {y + z} = \paren {x \times y} + \paren {x \times z}$
$\forall x, y, z \in \Z: \paren {y + z} \times x = \paren {y \times x} + \paren {z \times x}$

Corollary

The operation of multiplication on the set of integers $\Z$ is distributive over subtraction:

$\forall x, y, z \in \Z: x \times \paren {y - z} = \paren {x \times y} - \paren {x \times z}$
$\forall x, y, z \in \Z: \paren {y - z} \times x = \paren {y \times x} - \paren {z \times x}$

Proof

Let us define $\Z$ as in the formal definition of integers.

That is, $\Z$ is an inverse completion of $\N$.

From Natural Numbers form Commutative Semiring, we have that:

All elements of $\N$ are cancellable for addition
Addition and multiplication are commutative and associative on the natural numbers $\N$
Natural number multiplication is distributive over natural number addition.

The result follows from the Extension Theorem for Distributive Operations.

$\blacksquare$