Projections: Ordering Given a unital C*-algebra $1\in\mathcal{A}$.
Consider projections:
$$P^2=P=P^*\quad P'^2=P'=P'^*$$
Order them by:
$$P\leq P':\iff\sigma(\Delta P)\geq0\quad(\Delta P:=P'-P)$$

Then equivalently:
  $$P\leq P'\iff P=PP'=P'P\iff\Delta P^2=\Delta P=\Delta P^*$$

How can I check this?
(Operator algebraic proof?)

If $Q \ge P$, then
\begin{align}
      \|Px\|^{2} & =\|QPx\|^{2}+\|(I-Q)Px\|^{2} \\
            & = (PQPx,x)+\|(I-Q)Px\|^{2} \\
            & \ge (PPPx,x)+\|(I-Q)Px\|^{2} \\
            & = \|Px\|^{2}+\|(I-Q)Px\|^{2} \\
         \implies & (I-Q)Px=0.
\end{align}
