Equivalence of the lexicographic-composition de Bruijn constructions

Let $alt(n)$ denote the alternating sequence of $0$s and $1$s of length $n$ that ends with $0$. Let $\d835\d835\mathcal{L}_n$ be the de Bruijn sequence obtained from the lexicographic-composition construction, and let $\d835\d835\mathcal{LC}_n$ be the de Bruijn sequence returned by applying the de Bruijn successor $LC$ to $0alt(n-1)$. Equivalence of the lexicographic-composition constructions. The de Bruijn sequences $\d835\d835\mathcal{LC}_n$ and $\d835\d835\mathcal{L}_n$ are the same. This identifies the sequence produced by the successor-based construction with the sequence from the lexicographic-composition construction. The source presents the claim as an observed correspondence, but supplies no resolution beyond the stated assertion.