Product of Cardinals is Associative

Theorem
Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be cardinals.

Then:
 * $\mathbf a \left({\mathbf b \mathbf c}\right) = \left({\mathbf a \mathbf b}\right) \mathbf c$

where $\mathbf a \mathbf b$ denotes the product of $\mathbf a$ and $\mathbf b$.

Proof
Let $\mathbf a = \operatorname{Card} \left({A}\right)$, $\mathbf b = \operatorname{Card} \left({B}\right)$ and $\mathbf c = \operatorname{Card} \left({C}\right)$ for some sets $A$, $B$ and $C$.

We have by definition of product of cardinals that $\mathbf a \left({\mathbf b \mathbf c}\right)$ is the cardinal associated with $A \times \left({B \times C}\right)$.

Consider the mapping $f: A \times \left({B \times C}\right) \to \left({A \times B}\right) \times C$ defined as:
 * $\forall \left({a, \left({b, c}\right)}\right) \in A \times \left({B \times C}\right): f \left({a, \left({b, c}\right)}\right) = \left({\left({a, b}\right), c}\right)$

Let $\left({a_1, \left({b_1, c_1}\right)}\right), \left({a_2, \left({b_2, c_2}\right)}\right) \in A \times \left({B \times C}\right)$ such that:
 * $f \left({a_1, \left({b_1, c_1}\right)}\right) = f \left({a_2, \left({b_2, c_2}\right)}\right)$

That is:
 * $\left({\left({a_1, b_1}\right), c_1}\right) = \left({\left({a_2, b_2}\right), c_2}\right)$

It follows from Equality of Ordered Tuples that $a_1 = a_2, b_1 = b_2$ and $c_1 = c_2$.

That is:
 * $f \left({a_1, \left({b_1, c_1}\right)}\right) = f \left({a_2, \left({b_2, c_2}\right)}\right) \implies \left({a_1, \left({b_1, c_1}\right)}\right) = \left({a_2, \left({b_2, c_2}\right)}\right)$

and so $f$ is injective.

Now let $\left({\left({a, b}\right), c}\right) \in \left({A \times B}\right) \times C$.

By definition:
 * $\left({\left({a, b}\right), c}\right) = f \left({a, \left({b, c}\right)}\right)$

and so $f$ is a surjection.

So $f$ is both an injection and a surjection, and so by definition a bijection.

Thus a bijection has been established between $A \times \left({B \times C}\right)$ and $\left({A \times B}\right) \times C$.

It follows by definition that $A \times \left({B \times C}\right)$ and $\left({A \times B}\right) \times C$ are equivalent.

The result follows by definition of cardinal.