Euclidean space/Affine subspace/Perpendicular/Fact/Proof

We write ${\displaystyle {}P_{1}-P_{2}=u_{1}+u_{2}+u}$ with ${\displaystyle {}u_{1}\in U_{1}}$, ${\displaystyle {}u_{2}\in U_{2}}$, and ${\displaystyle {}u\in {\left(U_{1}+U_{2}\right)}^{\perp }}$; such a decomposition does always exist, ${\displaystyle {}u_{1},u_{2}}$ are not uniquely determined (in case ${\displaystyle {}U_{1}\cap U_{2}\neq 0}$), but ${\displaystyle {}u}$ is uniquely determined. We have

${\displaystyle {}P_{1}-u_{1}=P_{2}+u_{2}+u\,,}$

and ${\displaystyle {}Q_{1}:=P_{1}-u_{1}\in E_{1}}$, and ${\displaystyle {}Q_{2}:=P_{2}+u_{2}\in E_{2}}$. The distance between ${\displaystyle {}Q_{1}}$ and ${\displaystyle {}Q_{2}}$ is ${\displaystyle {}\Vert {u}\Vert }$. For arbitrary points ${\displaystyle {}R_{1}=Q_{1}+v_{1}\in E_{1}}$ and ${\displaystyle {}R_{2}=Q_{2}+v_{2}\in E_{2}}$ fulfilling ${\displaystyle {}v_{1}\in U_{1}}$ and ${\displaystyle {}v_{2}\in U_{2}}$, we have

${\displaystyle {}{\begin{aligned}d(R_{1},R_{2})^{2}&=\Vert {R_{1}-R_{2}}\Vert ^{2}\\&=\Vert {v_{1}-v_{2}+u}\Vert ^{2}\\&=\left\langle v_{1}-v_{2}+u,v_{1}-v_{2}+u\right\rangle \\&=\left\langle v_{1}-v_{2},v_{1}-v_{2}\right\rangle +\left\langle u,u\right\rangle \\&\geq \left\langle u,u\right\rangle ,\end{aligned}}}$

that is,

${\displaystyle {}d(R_{1},R_{2})\geq \Vert {u}\Vert \,.}$