real analysis – Can you check my proof of this equality?

Prove that

$$lVert x-yrVert^2+lVert x+y rVert^2 = 2 lVert x rVert^2+2 lVert y rVert^2 \ ,text{for all } x,y in mathbb{R}^p$$

My proof:

I will manipulate right-hand side of equation to make it equal to the LHS by adding and subtracting $2 lVert xrVert lVert yrVert $:
$$2 lVert x rVert^2+2 lVert y rVert^2 $$
$$2 lVert x rVert^2+2 lVert y rVert^2 + 2 lVert xrVert lVert yrVert – 2 lVert xrVert lVert yrVert ​$$
$$ lVert x rVert^2+ 2 lVert xrVert lVert yrVert + lVert y rVert^2 – 2 lVert xrVert lVert yrVert ​+ lVert x rVert^2 + 2 lVert xrVert lVert yrVert +lVert
yrVert^2 $$

$$ lVert x+y rVert ^2 + lVert x-y rVert ^2$$
Thus, RHS is equal to LHS.

The problem

It says that equality should hold for all $ x,y in mathbb{R}^P $. I am certain that this proof is correct for $p=1$ and $p=2$, but in higher dimensions I just don’t know. Can anyone highlight the weak side of the proof and give an advice on how to make it more solid.