# A difficulty in understanding the proof of boundary theorem in G&P.

The theorem and its proof is given below:

But I could not understand the last line in the proof in particular:

Why $$F^{-1}(Z)$$ is a compact one dimensional manifold with boundary? And why this leads to

(#$$\partial F^{-1}(Z)$$ = #$$f^{-1}(Z))$$?

Could anyone explain this for me please?

• $F^{-1}(Z)$ is a manifold because of transversality. – Lord Shark the Unknown Dec 9 '18 at 3:48
• By which theorem and why it is one dimensional?@LordSharktheUnknown – hopefully Dec 9 '18 at 3:57