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Let $M_1$ and $M_2$ be submanifolds of a manifold $M$. Assume that $M_1\cap M_2$ is also a submanifold of $M$ and that dim$(M_1)$+dim$(M_2)$-dim$(M_1\cap M_2)$=dim$(M) $. Is it true that $M_1$ and $M_2$ are transverse?

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No, take $M = \Bbb R^2$, $M_1 = \{(x,y) : y = 0 \}$ and $M_2 = \{(x,y) : y = x^2\}$.

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