Why "prove" the multi-value mean value theorem? The the theorem says the sup-norm of the change in a value of a mapping due a displacement is less than or equal to the product of the sup-norm of the displacement and the maximum value of the norm of the derivative matrix.  
That seems self evident.  The most by which the mapping could change would be if the largest component of displacement were the same component of the mapping with the greatest average rate of change.  The maximum possible value for that would be if its derivative of that component were constantly equal to the matrix norm of the derivative of the mapping.  Which is exactly what the theorem states.
 A: Look at intermediate value theorem. The intermediate value theorem is maybe the most self evidently theorem there is, it is so self evidently that for years the mathematicians community didn't thought about proving it but just took it as a fact. But after years the mathematicians understood that just because looks logical in every way it doesn't mean it is true, the intermediate value theorem is not as simple to prove as how easy it look.
The problem with self evidently things is that our logic is lacking, and sometimes the truth is contradiction to what we are thinking is the truth, so if we won't prove all of the things we are using we can waste a lot of time with working with theorems that may be false.
You need to understand 1 thing about this:$$\boxed{\text{self evidently}\ne\text{a fact}}$$
A: You need to prove also to verify under which hypothesis it is valid. In particular you need to assume that the domain of $f$ is path-connected.
Suppose $f$ is differentiable in $\overline{x}\in D$ then exists $M>0$ and $r>0$ such that...
A generalization of the mean value theorem?
