On a theorem concerning partially overlapping subpalindromes of a binary word (Q2070082)

The paper contains an alternative and maybe more natural proof of a technical theorem earlier proven and used by the authors in a previous paper [Discrete Appl. Math. 284, 434--443 (2020; Zbl 1448.68363)]: Let \(u \in \{1, 2\}^*\), let \(t, v \in 2^*\), and let \(p\) and \(q\) be subpalindromes of \(tu\) and \(uv\), respectively (meaning that they are palindromic scattered subwords of respective words). If the sum of lengths of \(p\) and \(q\) is at least twice the length of \(u\), then the number of \(1\)s in \(u\) is majorized by \(\frac{|tv|- 1}{|tv|} |tuv|_2\).