This work presents a formalized proof of modal completeness for G\"odel-L\"ob
provability logic (GL) in the HOL Light theorem prover. We describe the code we
developed, and discuss some details of our implementation, focusing on our
choices in structuring proofs which make essential use of the tools of HOL
Light and which differ in part from the standard strategies found in main
textbooks covering the topic in an informal setting. Moreover, we propose a
reflection on our own experience in using this specific theorem prover for this
formalization task, with an analysis of pros and cons of reasoning within and
about the formal system for GL we implemented in our code
