The lemma is an equivalence, and both directions are needed: one says the recursion never returns an entry no solution realizes, the other that it misses none.

The harder direction is the one that takes a solution compatible with $X$, in which $j$ is not selected, and re-assigns its jobs to the thresholds of $X_1$. The paper does this by sliding the machines' assignments up by one, which is an application of Hall's marriage theorem: the jobs that were on $j$ must move, and a threshold above them is free exactly because the jobs alive at any instant are few enough.

The hypothesis that the processing times are positive is used, and only in this direction. A job with $q_j = 0$ has an empty interval, conflicts with nothing, and may sit on the threshold $j$ below the cutoff $d_j$ — which is exactly the configuration the slide rules out. Soundness does not need it. The paper assumes positive processing times throughout.

The two branches are stated with the paper's own $X_1$ and $X_2$, including the cases in which $j_1$ or $j_2$ does not exist, where the threshold is dropped without replacement.