Can someone please explain in detail the following proof that affine spaces are isomorphic to vector spaces?
To clarify the notations the book has previously defined $R_{v}$ and $v_{x,y}$ as follows
An affine space is a set $\mathbb{A}^n$ with a collection of bijections $$\{R_{v}:\mathbb{A}^n\to\mathbb{A}^n:v\in\mathbb{R}^n\}$$ such that the following conditions are met (a) $R_{v+w} = R_{v} \circ R_{w}~\forall v,w\in \mathbb{R}^n$, (b) there exists unique $v_{x,y}\in \mathbb{R}^n$ such that $R_{v_{x,y}}(x)=y$ where $x,y\in\mathbb{A}^n$.
Let me explain why I am having a problem understanding the isomorphism. I understand isomorphism in the following way.
Given two sets with binary operations $(X,+_{X})$ and $(Y,+_{Y})$ (where $+_{X}:X\times X\to X$ and $+_{Y}:Y\times Y\to Y$) if one has bijection $f: X \to Y$ such that
$$f(x_{1}+_{X} x_{2}) = f(x_{1}) +_{Y} f(x_{2});~~x_{1},x_{2}\in X$$
and
$$f^{-1}(y_{1}+_{Y} y_{2}) = f^{-1}(y_{1}) +_{X} f(y_{2});~~y_{1},y_{2}\in Y$$
But in order to establish an isomorphism between $\mathbb{A}^n$ and $\mathbb{R}^n$ what is the binary operation on $\mathbb{A}^n$?
In the book's proof, what is the motivation for defining $\tilde{R}_{v}(x)$ like the way they define? Moreover, how does this definition imply $\tilde{v}_{x,y} = v_{\phi_{o}^{-1}(x),\phi_{o}^{-1}(y)}$? And how do all of these help with the isomorphism proof like the criteria I've written for sets $X$ and $Y$?
Moreover, I was scribbling some things like the following
Does this help in any way?