I am reading through Computational Geometry in C, 2nd Edition. In the book there is a proof for the existence of a triangulation that goes as follows:
If $n = 3$ the polygon is a triangle, and the theorem holds trivially. Let $n \ge 4$. Let $d = ab$ be a diagonal of $P$ as guaranteed by lemma 1.2.2. Because $d$ by definition only intersects $\partial{P}$ at its endpoints, it partitions P into two polygons each using $d$ as an edge, and each of fewer than $n$ vertices. The reason each has fewer vertices is that no vertices are added by this process, and clearly there is at least one vertex in each part in addition to $a$ and $b$. Applying the induction hypothesis completes this proof.
The bold portion is what has confused me. I am very familiar with induction - however when I have used induction it is always "explicit". For example, mentioning "now assume this holds for n, let's look at what happens to n+1" or something.
When the author states "apply the induction hypothesis" this, in my mind, is equivalent to "assume this holds for all $n$". I feel like it would be necessary for the author to prove the $n-1$th case here for the proof to be complete - however admittedly the author knows more about proofs than I do!
Can someone explain how this is a valid application of induction? I'd like to add it to my proof toolbox because it is very succinct and clear.