Given a Hilbert space $\mathcal{H}$.
Consider two orthogonal projections $P,Q$. Then: $$P\leq Q\implies\mathcal{R}(P)\subseteq\mathcal{R}(Q)$$
The ordering being induced by: $$T\geq0:\iff\mathcal{W}(T):=\{\langle T\hat{x},\hat{x}\rangle:\|\hat{x}\|=1\}\geq0$$
Can you give me a hint how to check that lemma?
For any orthogonal projection $P$, a vector $x$ is in the range of $P$ iff $\|Px\|=\|x\|$. The condition $P \le Q$ for orthogonal projections is equivalent to $\|Px\|\le \|Qx\|$. Because $\|Qx\|\le \|x\|$ it follows that if $x \in\mathcal{R}(P)$, then $x\in\mathcal{R}(Q)$.