# Integer Multiplication Distributes over Addition/Corollary

## Corollary to Integer Multiplication Distributes over Addition

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

 $\ds x \times \paren {y - z}$ $=$ $\ds x \times \paren {y + \paren {- z} }$ Definition of Integer Subtraction $\ds$ $=$ $\ds x \times y + x \times \paren {- z}$ $\ds$ $=$ $\ds x \times y + \paren {- \paren {x \times z} }$ Product with Ring Negative $\ds$ $=$ $\ds x \times y - x \times z$ Definition of Integer Subtraction

$\Box$

 $\ds \paren {y - z} \times x$ $=$ $\ds x \times \paren {y - z}$ Integer Multiplication is Commutative $\ds$ $=$ $\ds x \times y - x \times z$ from above $\ds$ $=$ $\ds y \times z - z \times x$ Integer Multiplication is Commutative

$\blacksquare$