Cardinal Product Distributes over Cardinal Sum

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

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

where:
 * $\mathbf a + \mathbf b$ denotes the sum of $\mathbf a$ and $\mathbf b$.
 * $\mathbf a \mathbf b$ 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 $C$.

Let $B$ and $C$ be pairwise disjoint, that is:
 * $B \cap C = \varnothing$

Then we can define:
 * $B \sqcup C := B \cup C$

where $B \sqcup C$ denotes the disjoint union of $B$ and $C$.

Then we have:
 * $\mathbf b + \mathbf c = \operatorname{Card} \left({B \sqcup C}\right) = \operatorname{Card} \left({B \cup C}\right)$

Then:

Then:

Then: