What is a Manifold? Lesson 12: Fiber Bundles - Formal Description

This is a long lesson, but it is not full of rigorous proofs, it is just a formal definition. Please let me know where the exposition is unclear. I din't quite get through the idea of the structure group of a fiber bundle fully, but I introduced it. The examples in the next lesson will help. I have repaired the title from a pervious version.