$\mathrm{L}^{\prime}$ is defined as the set of all strings $u$ such that there exists a string $v$ with $uv \in \mathrm{L}$ and $\text{length}(u) = \text{length}(v)$. Since $\mathrm{L}$ contains only strings of even length, both $u$ and $v$ must be of equal
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