Let $V_1,V_2\subset\mathbb{R}^n$ be two subspaces of dimension $d$. And $P_{V_1}, P_{V_2}$ are the orthogonal projections onto $V_1,V_2$ respectively. Let $V_1+V_2$ be the sum of these two subspaces and $P_{V_1+V_2}$ be the projection onto $V_1+V_2$. For an arbitrary $x\in\mathbb{R}^n$, do we have the following $$\|P_{V_1}x\|^2 + \|P_{V_2}x\|^2\geq \|P_{V_1+V_2}x\|^2?$$ Here $\|\cdot\|$ is the standard norm equipped in $\mathbb{R}^n$.
I tried using QR decomposition to find the orthonormal basis for $V_1+V_2$ but it doesn't work and the expression is very complicated. So I would like to ask for a simpler way to prove this or any counter example would be appreciated.
Let $n=2$, $x=(0,1)$, $V_1= \text{span}((1,0))$, and $V_2=\text{span}((1,1))$. Then
\begin{align} P_{V_1+V_2} x &= (0,1) \\ P_{V_1} x &= (0, 0) \\ P_{V_2} x &= (1/2, 1/2) \end{align}