Can order notation on its own imply:
$O(D(n)) + O(t(H)) - t(H) = O(D(n))$
My guess is that you cannot since the constant in the O(t(H)) would still exist after the subtraction if the constant is > 0.
Well, this is actually the case, but there are underlying factors. This equation appears in Fibonacci heap analysis in CLRS (518). The justification for this step comes from the underlying potential function. According to the authors, "we can scale up the units of potential to dominate the constant hidden in $O(t(H))$". I want to know how this happens, but don't really know how to ask this complicated question.