# Immersions

On page 19, O’Neill comments that the proof of Lemma 33 is a mild generalization of the proof of proposition 28. I think (2) $\iff$ (3) requires spelling out.

Let $\phi : M^m \longrightarrow N^n$ and let $\xi = (y^1,\ldots,y^n)$ be a co-ordinate system at $\phi (p)$. Let $\zeta = (x^1,\ldots,x^m)$ be a co-ordinate system at $p$. Then by (2) $\frac{\partial (y^i \circ \phi)}{\partial (x^j)}$ has rank $m$. Thus by exercise 7 and by re-arranging the co-ordinates if necessary, $(y^1 \circ \phi,\ldots,y^m \circ \phi)$ forms a co-ordinate system for $M^m$ on a neighbourhood $\cal{W}$ of $p$.

For the reverse, note that since $(y^1 \circ \phi,\ldots,y^m \circ \phi)$ is a co-ordinate system, by exercise 7, $\frac{\partial (y^i \circ \phi)}{\partial x^j}$ has rank $m$.