What is the correct way to describe a phrase that is a part of a sentence, taking into account the ordering of the words?

How do I write a criterion whereby it can be used to help someone decide that a name is considered a part of another name taking into account the ordering of the sequence of the name into account?

For example, consider the three names:

1. John Ronald Reuel Tolkien
2. Ronald Reuel
3. Reuel Ronald

This criterion would allow someone to understand that only name 1 & 2 would fulfill the criterion if the object is name 1.

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What I can currently think of is:

The Name must be an exact subset of another Name.

However, I am not confident that phrasing it this way would allow English speakers to immediately understand what I’m trying to say.

So I’m wondering, are there any phrases/words/ways out there to better describe what I want, or am I just being too self-conscious?