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.
Let x4 of type ι → ι → ι be given.
Let x5 of type ι → ι → ι be given.
Assume H1: x1 0.
Assume H2: ∀ x6 . x1 x6 ⟶ x1 (x3 x6).
Assume H3: ∀ x6 x7 . x1 x6 ⟶ x1 x7 ⟶ x1 (x5 x6 x7).
Assume H4: x2 0 = 0.
Assume H5: ∀ x6 . x1 x6 ⟶ x4 x6 0 = x6.
Assume H6: ∀ x6 . x1 x6 ⟶ x5 0 x6 = 0.
Assume H7: ∀ x6 . x1 x6 ⟶ x5 x6 0 = 0.
Let x6 of type ι be given.
Let x7 of type ι be given.
Assume H8: x1 x6.
Assume H9: x1 x7.
Apply CD_proj0_F with
x0,
x1,
x6,
λ x8 x9 . pair_tag x0 (x4 (x5 x9 (CD_proj0 x0 x1 x7)) (x2 (x5 (x3 (CD_proj1 x0 x1 x7)) (CD_proj1 x0 x1 x6)))) (x4 (x5 (CD_proj1 x0 x1 x7) x9) (x5 (CD_proj1 x0 x1 x6) (x3 (CD_proj0 x0 x1 x7)))) = x5 x6 x7 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H8.
Apply CD_proj1_F with
x0,
x1,
x6,
λ x8 x9 . pair_tag x0 (x4 (x5 x6 (CD_proj0 x0 x1 x7)) (x2 (x5 (x3 (CD_proj1 x0 x1 x7)) x9))) (x4 (x5 (CD_proj1 x0 x1 x7) x6) (x5 x9 (x3 (CD_proj0 x0 x1 x7)))) = x5 x6 x7 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H8.
Apply CD_proj0_F with
x0,
x1,
x7,
λ x8 x9 . pair_tag x0 (x4 (x5 x6 x9) (x2 (x5 (x3 (CD_proj1 x0 x1 x7)) 0))) (x4 (x5 (CD_proj1 x0 x1 x7) x6) (x5 0 (x3 x9))) = x5 x6 x7 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H9.
Apply CD_proj1_F with
x0,
x1,
x7,
λ x8 x9 . pair_tag x0 (x4 (x5 x6 x7) (x2 (x5 (x3 x9) 0))) (x4 (x5 x9 x6) (x5 0 (x3 x7))) = x5 x6 x7 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H9.
Apply H7 with
x3 0,
λ x8 x9 . pair_tag x0 (x4 (x5 x6 x7) (x2 x9)) (x4 (x5 0 x6) (x5 0 (x3 x7))) = x5 x6 x7 leaving 2 subgoals.
Apply H2 with
0.
The subproof is completed by applying H1.
Apply H6 with
x6,
λ x8 x9 . pair_tag x0 (x4 (x5 x6 x7) (x2 0)) (x4 x9 (x5 0 (x3 x7))) = x5 x6 x7 leaving 2 subgoals.