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.
Let x3 of type ι → ι be given.
Assume H1: ∀ x4 . x1 x4 ⟶ x1 (x2 x4).
Assume H2: ∀ x4 . x1 x4 ⟶ x1 (x3 x4).
Assume H3: ∀ x4 . x1 x4 ⟶ x2 (x2 x4) = x4.
Assume H4: ∀ x4 . x1 x4 ⟶ x3 (x3 x4) = x4.
Let x4 of type ι be given.
Apply CD_proj0R with
x0,
x1,
x4 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H5.
Apply CD_proj1R with
x0,
x1,
x4 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H5.
Apply H2 with
CD_proj0 x0 x1 x4.
The subproof is completed by applying L6.
Apply H1 with
CD_proj1 x0 x1 x4.
The subproof is completed by applying L7.
Apply CD_proj0_2 with
x0,
x1,
x3 (CD_proj0 x0 x1 x4),
x2 (CD_proj1 x0 x1 x4),
λ x5 x6 . pair_tag x0 (x3 x6) (x2 (CD_proj1 x0 x1 (pair_tag x0 (x3 (CD_proj0 x0 x1 x4)) (x2 (CD_proj1 x0 x1 x4))))) = x4 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L8.
The subproof is completed by applying L9.
Apply CD_proj1_2 with
x0,
x1,
x3 (CD_proj0 x0 x1 x4),
x2 (CD_proj1 x0 x1 x4),
λ x5 x6 . pair_tag x0 (x3 (x3 (CD_proj0 x0 x1 x4))) (x2 x6) = x4 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L8.
The subproof is completed by applying L9.
Apply H4 with
CD_proj0 x0 x1 x4,
λ x5 x6 . pair_tag x0 x6 (x2 (x2 (CD_proj1 x0 x1 x4))) = x4 leaving 2 subgoals.
The subproof is completed by applying L6.
Apply H3 with
CD_proj1 x0 x1 x4,
λ x5 x6 . pair_tag x0 (CD_proj0 x0 x1 x4) x6 = x4 leaving 2 subgoals.
The subproof is completed by applying L7.
Let x5 of type ι → ι → ο be given.
Apply CD_proj0proj1_eta with
x0,
x1,
x4,
λ x6 x7 . x5 x7 x6 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H5.