Proof: By Induction
(related to Proposition: Antiderivatives are Uniquely Defined Up to a Constant)
By hypothesis Let $I$ be a real interval and $f:I\to\mathbb R$ is a continuous function.
"$\Rightarrow$"
 Assume, $F,G:I\to\mathbb R$ are antiderivatives of $f.$
 By definition, this means for their derivatives that $F^\prime(x)=G^\prime(x)=f(x)$ for all $x\in I.$
 Therefore, for every $x\in I,$ $(F^\prime(x)G^\prime(x))=\frac d{dx}(FG)(x)=0$
 It follows from the derivative of a constant that $F$ and $G$ differ at most by a constant.
"$\Leftarrow$"
 Assume, $F(x)G(x)=c$ is constant for all $x\in I.$
 Then, $G^\prime(x)=\frac d{dx}(F(x)c)=F^\prime(x)=f(x)$ for all $x\in I.$
 It follows that $G$ and $F$ are antiderivatives of $f.$
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