The rest of the proof runs as before.
Although these proofs run along similar lines, there are subtle adjustments necessary to fit the argument to each new situation.
As t runs from 0 to 1, the point f(t) runs through the interval [a,b].
The problem one runs into, however, is that f need not be smooth.
But this obvious attack runs into a serious difficulty.
But if we argue as in (5), we run into the integral......, which is meaningless as it stands.
......where E runs over <runs through> the family B.