Does the current progress towards the next skill level carry over when training?

In Skyrim, you can level up your skilly 'manually' or by training. Each training session raises the level of the respective skill by 1. I'm wondering if this just raises my current skill level to the next full level with zero progression to the next level or if my current progress towards the next level carries over when training.

For example, let's say I have speech level 89 with the progress bar towards level 90 half full. I then use the master trainer to level up to level 90. I'm now level 90, but will my progress bar towards level 91 be completely empty or half full?

Side note: Same question, but for skill books?