# Natural Number Multiplication is Associative/Proof 1

## Theorem

The operation of multiplication on the set of natural numbers $\N$ is associative:

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

## Proof

From Index Laws for Semigroup: Product of Indices we have:

$+^{z \times y} x = +^z \left({+^y x}\right)$

By definition of multiplication, this amounts to:

$x \times \left({z \times y}\right) = \left({x \times y}\right) \times z$

From Natural Number Multiplication is Commutative, we have:

$x \times \left({z \times y}\right) = x \times \left({y \times z}\right)$

$\blacksquare$