@AlexDegtyarev You mentioned On the other hand, any positive distance can be made smaller and you gave a reference to Milnor's Morse. But after skimming through the proof of the Lefschetz theorem, there is no mention of the distance function having a positive index. There is a bound though, and that is what is used in that book. Also in the answer there is no mention of the fact that $V$ and $V'$ have complementary dimension.