The greedy is given as a relation between the set before a step and the set after it, not as a function. Both rules leave a choice — which job with the largest due date to drop — and a relation covers every way of resolving it at once, so the theorem is about the rule and not about one implementation of it.

The domination order is not the paper's, and the paper's will not do. Section 6.1 defines: $S$ dominates $S'$ if $|S| > |S'|$, or $|S| = |S'|$ and the $i$-th largest due date in $S$ is not greater than the $i$-th largest in $S'$. The first disjunct throws away all information about the due dates whenever the cardinalities differ, and the induction needs it exactly there: in the step that drops a job, the set compared against has one element fewer, so the induction hypothesis says only that the greedy's set is larger, and "the job removed is the one with the largest due date" has nothing left to act on.

Dropping the disjunct repairs it. Counting, for each threshold $t$, how many due dates of a set lie at or below $t$ gives an order that is the paper's pointwise condition when the cardinalities agree, that implies $|S'| \le |S|$, and that every greedy step preserves — which is what the induction needs. Stating it by counting rather than by listing the sorted due dates also turns every step of the argument into arithmetic.