# Use of Implicit Function Theorem to provide examples of Manifolds

Suppose we know Definition 1 and Theorem 1, and want to prove Theorem 2 given below.

Definition 1: A subset $M$ of $R^n$ is called an k-dimensional manifold (in $R^n$) if for each point $x\in M$ the following condition is satisfied:

(A) There is an open set $U$ containing $x$, an open set V$\subset$$R^n, and a diffeomorphism h:U\rightarrow V such that$$h(U\cap M)=V\cap (R^k \times \lbrace0\rbrace)=\lbrace y\in V:y^{k+1}=\cdot\cdot\cdot\cdot\cdot\cdot=y^n=0\rbrace$$Theorem 1: Let f:R^n \rightarrow R^p be continuously differentiable in an open set containing a, where p\le n. If f(a)=0 and the p\times n matrix (D_jf^i(a)) has rank p, then there is an open set A\subset R^n containing a and a differentiable function h:A\rightarrow R^n with differentiable inverse such that$$f\circ h(x^1,....,x^n)=(x^{n-p+1},.....x^n)$$Theorem 2: Let$A\subset R^n$be open and let$g:A\rightarrow R^p$be a differentiable function such that$g\prime(x)$has rank$p$whenever$g(x)=0$. Then$g^{-1}(0)$is an$n-p$dimensional manifold in$R^n.\$

How to prove theorem 2 using Theorem 1 and Definition 1?

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