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?