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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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