Proving Indefinite Integral Property 2

We want to prove: $\forall c\in \mathbb{R},\int cf(x)dx=c\int f(x)dx$

Proof

1. Let $c\in \mathbb{R}$ be arbitrary
2. Proving $\Leftarrow$
   1. Suppose $c\int f(x)dx$ exists
   2. $=c(F(x)+C)$ by definition of Indefinite Integral
   3. $=cF(x)+cC$
   4. Note that $cC$ is a constant, thus, $\exists cF(x)+C$ which is the antiderivative for $f$
   5. Note that $(cF(x){)}^{\prime }=cf(x)$
   6. Thus, by definition of Indefinite Integral, $\int cf(x)dx$ exists