If $M$ in an $m$-dimensional embedded submanifold of $\mathbf{R}^2$, then $TM$ is an embedded sub manifold of $T\mathbf{R}^n$ because we have the normal bundle $NM$ is an embedded sub manifold and $TM$ is its orthogonal complement.
My question is, is there a direct way to prove that $TM$ is embedded using slice charts? I tried writing out the slice charts for $M$ but then not sure how to get to slice charts of $TM$.