Niddah, Chapter 6, Mishnah 11. This chapter gathers a long list of general rules, all of them cast in a single mold: whatever is true in one case is necessarily true in a second, though the second does not carry back to the first. Having applied that formula to matters of tumah and taharah and to the simanim of kosher creatures, the Mishnah now applies it to berachos.
Here is the principle itself. "Kol hataun brocha l'achorov", whatever calls for a blessing once it has been completed, is likewise "taun brocha l'fanov", something over which a blessing is said at the outset. Eating illustrates the point plainly: since a person owes a blessing when his meal is finished, it goes without saying that he owed one when he sat down to begin it.
But the rule does not run in the other direction: "Yesh shetaun brocha l'fanov v'ein taun brocha l'achorov", there are things over which we recite a brocha beforehand and nothing at all afterward.
What sort of things? Anything that is a dvar mitzvah. We take the lulav and recite a brocha before performing the mitzvah, and once the mitzvah has been performed we say nothing further. The same holds for blowing shofar, for affixing a mezuzah, for putting on tefillin, and for the other mitzvos like them. Every one of them opens with a brocha, and none of them closes with one.
So the connection between the two berachos is one sided. A brocha after the act proves that there was a brocha before it, while a brocha before the act proves nothing about what follows.