Suppose two intersecting smooth manifolds which are both subset of $\mathbb{R}^2$, and their tangent spaces on points of the intersecting parts doesn't coincident. Then is this intersecting part a 1manifold? If yes, why?
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The answer is Implicit function theorem, but I did not get what was the question. 


Let $X$ and $Y$ be two (nice) submanifolds of $Z,$ intersecting transversely, then the resulting intersection is a manifold of $\text{codim} (X \cap Y)= \text{codim}(X) + \text{codim}(Y).$ 

