My book states, that if we apply the "mean value theorem in integral form" on $f'$, then follows: $\int_a^b{f'} = (b-a)f'(c)$. If we compare this with the "canonical mean value theorem": $f(b) - f(a) = (b-a)f'(c)$, then this suggests that $\int_a^b{f'} = f(b) - f(a)$. It also states that although the former suggests the latter, it doesn't prove the latter.
My question is why isn't this a proof?
