I would like to get some explanations for this fact. (The plane is $\mathbb{R}^2$)
So, I've been drawing on a grid to see what a composition of $R(a,\alpha) R(b,-\alpha)$ (but, $a \neq b$) gives us.
And I could convince myself this is indeed a translation.
However, I am stuck when showing this with mathematical terms.
So far, I've shown that $R(a,\alpha)R(b,-\alpha)=R_pR_lR_mR_q=R_p T_x R_q$ where $R_p, R_q$ are reflections in line $p,q$ and $T_x$ is a translation that is given by $R_lR_m$ with $T_x(v)=v+x$ (becasue $l,m$ are parallel.)
I've also found that $R_pT_xR_q=T_{x+y}$ where $y$ is a vector s.t the direction of it is perpendicular to $p,q$ and the length of it is the distance between parallel lines $p,q$
How can I convert these facts to formal proof?