Euclidean space/Distance to subspace/Inner product/Fact

Let ${}V$ be a Euclidean vector space, ${}U\subseteq V$ a linear subspace, and ${}v\in V$. Let ${}u_{1},\ldots ,u_{m}$ denote an orthonormal basis of ${}U$.

Then

${}d(v,U)^{2}=\Vert {v}\Vert ^{2}-\sum _{i=1}^{m}\left\langle v,u_{i}\right\rangle ^{2}\,.$

Proof, Write another proof