If $f$ belongs to $M^{+} $ and $c \ge 0$ then $cf$ belongs to $M^{+}$ and $ \int cf = c\int f$

If $ f$ belongs to $ M^{+} $ and $ c \ge 0$ then $ cf$ belongs to $ M^{+}$ and $ \int cf = c\int f$ .

I need to proove that, using the following observation:

if $ f\in M^{+}$ and $ c>0 $ , then the mapping $ \varphi \rightarrow \psi = c\varphi$ is a one-toone mapping between simple function $ \varphi \in M^{+}$ with $ \varphi \le f $ and simple functions $ \varphi$ in $ M^{+} $ with $ \psi \le cf $ .

I know that this question is already answer here:One-to-one mapping of simple functions $ \phi \to \psi = c\,\phi$ implies $ \int cf\,d\mu = c \int f\,d\mu$ ?

But I can’t follow the verbal explanation.

My original idea was to proove $ $ c \int f \le \int cf \le c\int f $ $ But I can’t… some idea?