I'm trying to understand this fact that looks very intuitive drawing, but I'm completely stuck trying to prove it formally.
Let $M$ a smoot manifold of dimension $n$ without boundary. Then $M \times [0,1]$ is a smooth manifold with boundary $\partial(M \times [0,1]) = M\times\{0\} \sqcup M\times\{1\}$. In addition, if $M \times [0,1]$ is oriented the orientation induced on $M\times\{0\}$ and $M\times\{1\}$ are opposite (where I'm identifying $M\times\{t\}$ with $M$).
Any kind of help will be appreciated.