Include proof in bsc thesis that I couldn't do fully myself
I don't see an issue here for an undergraduate thesis. If the "important paper" is new and the work overlapped with your own, you should be able to get credit for your work independently of the other. If it is a classic paper that you should have known about, but didn't, then you may have a more serious issue and a reason to re-write. 

But citing the other work, just as in your example, seems right, including the main conclusions you draw from the paper. 

But the professor to whom you are responsible (maybe not just the assistant) is the best source for an answer to this question. 

If there is any serious issue raised by the professor, then you need, first, to think about what if anything is unique about your solution. It may be that a relatively minor re-write will do the trick if you just stress what is new/unique about it. 

I'd also suggest some perspective. The fact that, as an undergraduate, you can "follow" but not "replicate" the work in another paper may be good or bad depending on the other paper. We normally expect that students at this level are students and thus can't know or do everything. Even experts can struggle with some deep ideas and intricate proofs. 

I'll note that it would be a more serious issue for a doctoral dissertation. There is a fairly recent case, two people known to me, who had their doctorates delayed for a year because they each did a piece of nearly identical work without knowledge of the other. They didn't need to re-do any work, but it took a year for the (Computer Science) community to determine that this was just parallel work and no collusion nor plagiarism was involved. Sticky, but both are now highly regarded.