Let x0 of type ι be given.
Let x1 of type ι → ο be given.
Assume H0:
∀ x2 . x1 x2 ⟶ ∀ x3 . x3 ∈ x2 ⟶ nIn x0 x3.
Let x2 of type ι → ι be given.
Assume H1: ∀ x3 . x1 x3 ⟶ x1 (x2 x3).
Assume H2: ∀ x3 . x1 x3 ⟶ x2 (x2 x3) = x3.
Let x3 of type ι be given.
Apply CD_proj0R with
x0,
x1,
x3 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H3.
Apply CD_proj1R with
x0,
x1,
x3 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H3.
Apply H1 with
CD_proj0 x0 x1 x3.
The subproof is completed by applying L4.
Apply H1 with
CD_proj1 x0 x1 x3.
The subproof is completed by applying L5.
Apply CD_proj0_2 with
x0,
x1,
x2 (CD_proj0 x0 x1 x3),
x2 (CD_proj1 x0 x1 x3),
λ x4 x5 . pair_tag x0 (x2 x5) (x2 (CD_proj1 x0 x1 (pair_tag x0 (x2 (CD_proj0 x0 x1 x3)) (x2 (CD_proj1 x0 x1 x3))))) = x3 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L6.
The subproof is completed by applying L7.
Apply CD_proj1_2 with
x0,
x1,
x2 (CD_proj0 x0 x1 x3),
x2 (CD_proj1 x0 x1 x3),
λ x4 x5 . pair_tag x0 (x2 (x2 (CD_proj0 x0 x1 x3))) (x2 x5) = x3 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L6.
The subproof is completed by applying L7.
Apply H2 with
CD_proj0 x0 x1 x3,
λ x4 x5 . pair_tag x0 x5 (x2 (x2 (CD_proj1 x0 x1 x3))) = x3 leaving 2 subgoals.
The subproof is completed by applying L4.
Apply H2 with
CD_proj1 x0 x1 x3,
λ x4 x5 . pair_tag x0 (CD_proj0 x0 x1 x3) x5 = x3 leaving 2 subgoals.
The subproof is completed by applying L5.
Let x4 of type ι → ι → ο be given.
Apply CD_proj0proj1_eta with
x0,
x1,
x3,
λ x5 x6 . x4 x6 x5 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H3.