Lemma 16 ([Crossing the zones]). If A⊆Z⊆E₀ , with A Lower and Z Upper, then there is an admissible B satisfying A⊆B⊆Z .

Lemma 15. The Upper zone, Lower zone, and admissible sets partition 2^E₀ . The Upper zone is upward closed, the Lower zone is downward closed, and all three statuses are invariant under symmetric differences in K₀ . Complementation …

Lemma 7 ([Rank-raising thinning]). Let s⊆X⊆D , with s small and X large, and let γ< c . There is a large set u such that s⊆u⊆X, ρ(u)>γ.

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.

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Lemma 16 ([Crossing the zones]). If A⊆Z⊆E0 , with A Lower and Z Upper, then there is an admissible B satisfying A⊆B⊆Z .

Lemma 15. The Upper zone, Lower zone, and admissible sets partition 2E0 . The Upper zone is upward closed, the Lower zone is downward closed, and all three statuses are invariant under symmetric differences in K0 . Complementation …

Lemma 7 ([Rank-raising thinning]). Let s⊆X⊆D , with s small and X large, and let γ< c . There is a large set u such that s⊆u⊆X, ρ(u)>γ.

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Lemma 16 ([Crossing the zones]). If A⊆Z⊆E₀ , with A Lower and Z Upper, then there is an admissible B satisfying A⊆B⊆Z .

Lemma 15. The Upper zone, Lower zone, and admissible sets partition 2^E₀ . The Upper zone is upward closed, the Lower zone is downward closed, and all three statuses are invariant under symmetric differences in K₀ . Complementation …