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The length of a longest repeated substring equals the maximum longest-common-prefix length over adjacent suffixes in lexicographic order. Suffixes beginning with a fixed string form a consecutive interval: a string outside that string prefix software class differs earlier or ends sooner, putting it strictly to one side. Two occurrences of a longest repeated substring therefore bound an interval in which every adjacent pair shares that string prefix. At least one adjacent pair attains its length, and no pair can exceed the global maximum. This proves the result even when occurrences overlap; a text with fewer than two suffixes needs a separate empty-result convention.

All suffixes beginning with a fixed string form an interval in lexicographic order. To see this, suppose two suffixes start with a string $u$, while a string suffix between them fails to start with $u$. At its first differing character it must be either smaller than both or larger than both; if it ends before $u$ does, it is smaller than both. Either possibility contradicts its position between them.

Now take a longest repeated substring $u$. Its two occurrences are prefixes of two distinct suffixes. Every string suffix between them also begins with $u$, so at least one adjacent pair in that interval has a common string prefix of length at least $|u|$. A greater length would contradict maximality of $u$. Therefore the maximum common-prefix length over adjacent sorted suffixes equals the longest repeated-substring length. If the maximum is zero, any adjacent pair still attains it. The argument does not assume nonoverlapping occurrences.