I came across the following statements while reading:
Consider two affine subspaces $L = x_0 + U$ and $L = x'_0 + U'$ of a vector space $V$.
Then, $L \subseteq L'$ iff $U \subseteq U'$ and $x_0 −x'_0 \in U'$.
My original question: why is the condition $x_0 −x'_0 \in U'$ required for $L$ to be a subspace of $L'$?
I found an answer here, but I'm not sure I understand this part:
"Then, let us consider an element $z\in U_1$. We know by definition that $x_1+z$ belongs to $L_1\subset L_2=x_2+U_2$. It follows that $z\in (x_2-x_1)+U_2=U_2$ because we already know that $x_1-x_2 \in U_2$."
how does it follow that $z\in (x_2-x_1)+U_2$ (and how is this equal to $U_2$)?