# Formalizing a proof using Germs to define a linear and injective map of the Algebraic Tangent of a manifold

I am trying to show that for X being an n-dimensional manifold and Y a k-dimensional manifold, U an open set and both $Y,U \subset X, p\in U$ there is a natural and well defined and injective map $$i_{*}^{alg}:T_{p}^{alg}Y \rightarrow T_{p}^{alg}X \text{ is defined by } i_{*}^{alg}(v)([f,U]) \equiv v([f|_{U \cap Y}, U \cap Y])$$

Now I am guessing that wehat I am needing to show is that $f|_{U \cap Y}: U \cap Y \rightarrow \mathbb{R}$ is differentiable, that $(f|_{U \cap Y}, U \cap Y) \text{ defines the germ } [f|_{U \cap Y}, U \cap Y]$ of a differentiable function on $Y \text{ near } p, \text{where } f:U \rightarrow \mathbb{R}$ is a differentiable function. But this is just the first step. Could I say that this is true because $g:Y \rightarrow \mathbb{R}^k = \mathbb{R}^{k-1} \times \mathbb{R}$ and so $U \cap Y$ is essentially $f \cap g: U \cap Y \rightarrow \mathbb{R} \cap (\mathbb{R}^{k-1} \times \mathbb{R}) = \mathbb{R}?$

Now to show the definition of $i_{*}^{alg}$ works, we would have to show that $i_{*}^{alg} \circ (v \circ([f,U]))$ is as defined above, but that $(v \circ([f,U])) \in T_{p}^{alg}Y$. Is that the key? Any thoughts on how I could show this?