Prove
Lemma 7 ([Rank-raising thinning]). Let s⊆X⊆D , with s small and X large, and…Lemma 15. The Upper zone, Lower zone, and admissible sets partition 2^ E_0 .…Proposition
The sets s=A_L and X=Z_L are respectively small and large, and s⊆X . We keep B_R=A_R , so write y=D A_R . If y is large, Lemma 7 ([Rank-raising thinning]) gives a large u between s and X with ρ(u)>ρ(y) ; take B_L=u . If y is small, apply the same Lemma between the small set D X and the large set D s to obtain a large v with ρ(v)>ρ(D y) . Taking B_L=D v gives the second line of. Both constructions respect the two endpoint inclusions.