# What is wrong with this "proof" of $coNP = NP$? [duplicate]

I am wondering why the following "proof" of $$coNP = NP$$ does not work:

$$\subseteq:$$Let $$L$$ be a language in $$coNP$$, that means there is a non-deterministic Turing Machine $$M$$ that decides the complement of $$L$$, denoted by $$\overline{L}$$, in polynomial time. Then the Turing Machine $$N$$ that decides $$L$$ in polynomial time by reversing the result of running $$M$$ on $$\overline{L}$$.

$$\supseteq:$$ Reverse the above arguments.

Remark: I know that this "proof" can not be correct in this way, since $$coNP = NP$$ is still an open problem and it is certainly not that easy. But I do not understand where the reasoning above goes wrong. Could you please explain this to me?