On changing cofinality of partially ordered sets

Definition 4.18 Let 〈η, T (⃗p)〉, 〈η ′ , T ′ (⃗p ′ )〉 ∈ P . Set 〈η, T (⃗p)〉 ≤ 〈η ′ , T ′ (⃗p ′ )〉 iff
1. η ′ ∈ T (⃗p),
2. η ′ is an end extension of η,
3. T ′ (⃗p ′ ) ⊆ T (⃗p),
4. ⃗p η ′ ≤ ⃗p ′ .
Definition 4.19 Let 〈η, T 〉, 〈η ′ , T ′ 〉 ∈ P . Set 〈η, T 〉 ≤ ∗ 〈η ′ , T ′ 〉 iff
1. 〈η, T 〉 ≤ 〈η ′ , T ′ 〉,
2. η = η ′ .
Lemma 4.20 Let 〈p(0), p(1), ..., p(n), ... | n < ω〉 be a good sequence and q ′ (0) ≥ p(0). Then
there are 〈q(0), q(1), ..., q(n), ... | n < ω〉 such that
1. q(0) ≥ q ′ (0),
2. 〈q(0), q(1), ..., q(n), ... | n < ω〉 is a good sequence,
3. for each n < ω, q(n) ≥ p(n),
Proof. Combine j ≤n (q ′ (0)) with p(n), for each n < ω, and turn the results into separated
conditions. Let us argue that (6) of Definition 4.12 can be easily satisfied. Thus, we need
to take care of α’s in a 0 (q ′ (0)) \ a 0 (p(0)) such that j ≤l (α) does not appear in a n(α) (p(l)), for
1 ≤ l < ω. Given such α, we consider k’s above n(α). Then j ≤k (α) ∉ a n(α) (p(k)). But then
we are free to set
〈j ≤k (α), κ k 〉 ∉ ⊤ q(k),n(α) .
□
Denote the result for each n < ω, by q(n). By the construction we have q(n) ≥ p(n).
The following is a slightly more general statement with a similar proof.
Lemma 4.21 Let 〈p(0), p(1), ..., p(n), ... | n < ω〉 be a good sequence, l < ω and q ′ (m) ≥
p(m), for every m ≤ l. Then there are 〈q(0), ..., q(n), ... | n < ω〉 such that
1. q(0) = q ′ (0),
2. q(m) ≥ q ′ (m), for every m ≤ l,
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