If $L$ is a affine subspace of $E_n$ and $b\in L$, then the set $T_{-b}(L)\subset E_n$, where $T_{-b}(x):=x-b, x\in L$, is a vector subspace of $E_n$.
I saw a demonstration that use a "trick" and prove first that $\alpha \,v \in L$, and next that $v_1+v_2 \in L$.
My question is about another demonstration wich consider that $z:=\alpha\, T_b(y_1)+\beta\, T_b(y_2)+(1-\alpha-\beta)b \in L$, because $T_b(y_1)$ and $T_b(y_2)$ belong to $L$ for $y_i\in T_{-b}(L)$.
I don't understand why $z \in L$. Can someone help?
Thanks
p.s.: $E_n$ is the usual euclidean vector space, $v_i\in E_n$ and $\alpha,\beta$ are scalars