User:Arbo/Sandbox.

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Theorem
Every smooth $m$-dimensional manifold can be immersed in Euclidean $\left({2m-1}\right)$-space.

Proof
We will first show that any compact manifold can be embedded (immersed) in some $\R^N$, $N \gt\gt 0$. This argument can be extended to the case of arbitrary manifolds.

Proposition 1
For every injective Function $f\colon M \rightarrow \R^n$ there exists a linear Transformation $\lambda \colon \R^n \rightarrow \R^{n-1}$, such that $\lambda \f$ is injective, provided that $n \gt 2m-1$.

Then by definition:
 * [Definition 1 of definiend]



Thus $...$ is a [definiend] by definition 2.

$(2)$ implies $(1)$
Let $...$ be a [definiend] by definition 2.

Then by definition:
 * [Definition 2 of definiend]



Thus $...$ is a [definiend] by definition 1.

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