Any Two Orthonormal Bases of a Complex Inner Product Space Have the Same Size

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a \reftext{def:complex-inner-product-space-2026a}{complex inner product space}, let $m$ and $n$ be \reftext{def:natural-numbers-2026a}{natural numbers}, and let $e\in V^{n}$ and $f\in V^{m}$ be \reftext{def:finite-tuple-power-2026a}{tuples} in $V$ that are both \reftext{def:orthonormal-basis-2026b}{orthonormal bases} of $V$. Then $m=n$.