Where is \(u'\text{?}\) It cannot be a vertex of \(P\text{,}\) because if it was, there would be two distinct paths from \(u\) to \(u'\text{:}\) the edge between them, and the first part of \(P\) (up to \(u'\)). But \(u'\) also cannot be outside of \(P\text{,}\) for if it was, there would be a path from \(u'\) to \(v\) that was longer than \(P\text{,}\) which has longest possible length.
in-context