Cartesian Product Preserves Cardinality

Theorem
Let $R$, $S$, and $T$ be sets.

Suppose that $S$ is equivalent to $T$.

Then:


 * $R \times S \sim R \times T$


 * $S \times R \sim T \times R$

Proof
Since $S$ and $T$ are equivalent, there exists a bijection $f: S \to T$.

Let $g: T \to S$ be the inverse of $f$; its existence is assured by Bijection iff Left and Right Inverse.

Define $\hat f: R \times S \to R \times T$ by:


 * $\hat f \left({r, s}\right) := \left({r, f \left({s}\right)}\right)$

Next, define $\hat g: R \times T \to R \times S$ by:


 * $\hat g \left({r, t}\right) := \left({r, g \left({t}\right)}\right)$

Then, for all $\left({r, s}\right) \in R \times S$:

Similarly, for all $\left({r, t}\right) \in R \times T$:

Thus, it follows that $\hat g$ is a inverse for $\hat f$.

From Bijection iff Left and Right Inverse, we conclude $\hat f$ is a bijection.

Hence $R \times S \sim R \times T$, by definition of set equivalence.

Mutatis mutandis, the other relation also follows.