Differentiable function/Mean value theorem/Quotient version/Fact

Mean value theorem (quotient version)

Let ${}b>a$, and suppose that

$f,g\colon [a,b]\longrightarrow \mathbb {R}$

are continuous functions which are differentiable on ${}]a,b[$ and such that

${}g'(x)\neq 0\,$

for all ${}x\in {]a,b[}$.

Then

${}g(b)\neq g(a)$, and there exists some ${}c\in {]a,b[}$ such that

${}{\frac {f(b)-f(a)}{g(b)-g(a)}}={\frac {f'(c)}{g'(c)}}\,.$

Proof, Write another proof