On changing cofinality of partially ordered sets

Proof. Let σ be a statement and 〈〈p, 〈η, T 〉〉 ∈ Q ∗ P . Suppose for simplicity that η is the
∼ ∼
empty sequence.
Let ⃗p = 〈p(0), p(1), ..., p(n), ...〉 be a good sequence with T (⃗p) = T and p(0) ≥ p.
For each ν < κ 1 we define by induction a sequence ⃗r(ν) = 〈r(1, ν), r(2, ν), ..., r(n, ν), ... | 1 ≤
n < ω〉 by induction as follows.
Suppose that for each ν ′ < ν the sequence ⃗r(ν ′ ) is defined. Define ⃗r(ν).
Case 1. ν = ν ′ + 1.
Consider first the sequence ⃗r ∗ (ν ′ ) = 〈r ∗ (1, ν ′ ), ..., r ∗ (n, ν ′ )... | 1 ≤ n < ω〉 which is obtained
from ⃗r(ν ′ ) as follows:
• for each 1 ≤ k < ω, let (r ∗ (k, ν ′ )) 0 = j ′′ ≤k p(k) 0 ;
• for each k, m < ω, 1 ≤ k, m let (r ∗ (k, ν ′ )) m = (r(k, ν ′ )) m .
Set
⃗r(ν) = ⃗r ∗ (ν ′ ),
unless there is a good sequence 〈r ′ (n) | n < ω〉 such that
• 〈r ′ (n) | n < ω〉 ≥ ⃗r ∗ (ν ′ ),
• r ′ (1) 〈ν〉 ‖ Q
(〈ν, ∼
T (〈r ′ (2) 〈ν〉 , ..., r ′ (n) 〈ν〉 , ... | 2 ≤ n < ω〉)〉‖σ).
In this case let ⃗r(ν) be such a sequence.
Case 2. ν is a limit ordinal.
Then we define first a sequence ⃗r ∗ (ν) as follows:
• for each 1 ≤ k < ω, let (r ∗ (k, ν)) 0 = j ′′ ≤k p(k) 0 ;
• for each k, m < ω, 1 ≤ k, m let (r ∗ (k, ν)) m = ⋃ ν ′