Let $I$ be an D1247: Inner product-normed vector space such that

(i)	$\langle \cdot, \cdot \rangle$ is the D34: Inner product in $I$
(ii)	$\Vert \cdot \Vert$ is the D504: Inner product norm in $I$
(iii)	$x, y \in I$ are each a D1129: Vector in $I$

We have \begin{equation} \begin{split} \Vert x + y \Vert^2 & = \left( \sqrt{\langle x + y, x + y \rangle} \right)^2 \\ & = \langle x + y, x + y \rangle \\ & = \langle x, x \rangle + 2 \langle x, y \rangle + \langle y, y \rangle \\ & = \Vert x \Vert^2 + 2 \langle x, y \rangle + \Vert y \Vert^2 \end{split} \end{equation} and \begin{equation} \begin{split} \Vert x - y \Vert^2 & = \left( \sqrt{\langle x - y, x - y \rangle} \right)^2 \\ & = \langle x - y, x - y \rangle \\ & = \langle x, x \rangle - 2 \langle x, y \rangle + \langle y, y \rangle \\ & = \Vert x \Vert^2 - 2 \langle x, y \rangle + \Vert y \Vert^2 \end{split} \end{equation} Adding these equations together, we then conclude \begin{equation} \begin{split} \Vert x + y \Vert^2 + \Vert x - y \Vert^2 & = \Vert x \Vert^2 + 2 \langle x, y \rangle + \Vert y \Vert^2 + \Vert x \Vert^2 - 2 \langle x, y \rangle + \Vert y \Vert^2 \\ & = 2 \Vert x \Vert^2 + 2 \langle x, y \rangle + 2 \Vert y \Vert^2 - 2 \langle x, y \rangle \\ & = 2 \Vert x \Vert^2 + 2 \Vert y \Vert^2 \end{split} \end{equation} $\square$