He’s proving the existence of authority (with a definition thats wide/encompasses the libertarian framework).
He’s not using that definition anywhere in his article.
If you know think about going for the “but Engel’s definition is broader, therefore, his argument is still valid” boy oh boy I suggest you study logic. That’s not how widening and narrowing works.
Say, cooks. They say: “These things are fruits, and with them we can make fruit salads”. Botanists say “These things are fruit, our category is wider, it includes tomatoes, therefore, you can make fruit salad with tomatoes”.