rule situation

Question
This just happened in my son's game last night.  His team was in the field.  Bases loaded, two outs, pop fly on the infield and it is not caught. But the runner on second apparently forgot there were two outs, and had only gone halfway. So the third baseman picks up the ball and tags third. Our view: it was a force out and the run does not count even though the runner on third crossed home before the put out at third. The umpire ruled that the run counted because the runner crossed the plate before the put out AND because the put out was made AFTER the batter reached first. Reading the rules of baseball, I don't understand the ruling at all. Under Rule 4.09 the run clearly does not count if the third out is a force out. Under the definition of "Force out," I don't see how this was not. If it wasn't a force out, then how could he even have been out? All I can imagine is that somehow the act of the batter crossing first before the put out at third removed the put out from the definition of a "force out" (does it become two runners one base?), but that's certainly not apparent from the language of the definition. The umpire would/could not explain it.

Answer
Your view is correct.  Since there was no unoccupied base preceding the runner who started at second, he is forced to advance forward to third base.  If third base is touched with the ball prior to him reaching 3rd base, he is out on a force and the run scoring from third would be disallowed.  The key with the force rule comes down to whether or not there is an open base preceding the runner (the batter is automatically "taking" first in all situations).  Unless someone behind a runner gets out he is forced to advance.