These results now look stable enough for publication. We begin by fixing standard terminology and notation regarding posets. If there is no longest length, then the height is considered to be infinity. We use N for the set of all nonnegative integers. Here R is the set of all real numbers.
Finite range means that the range is a finite set.
And here is a sharper form. First we begin with two semifinite forms. The hypotheses are infinite but the conclusion is finite. Here we will not need the parameter r. And here is a sharper form.