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 → ι be given.
Assume H1: x1 0.
Assume H2: x2 0 0 = 0.
Let x3 of type ι be given.
Let x4 of type ι be given.
Assume H3: x1 x3.
Assume H4: x1 x4.
Apply CD_proj0_F with
x0,
x1,
x3,
λ x5 x6 . pair_tag x0 (x2 x6 (CD_proj0 x0 x1 x4)) (x2 (CD_proj1 x0 x1 x3) (CD_proj1 x0 x1 x4)) = x2 x3 x4 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
Apply CD_proj1_F with
x0,
x1,
x3,
λ x5 x6 . pair_tag x0 (x2 x3 (CD_proj0 x0 x1 x4)) (x2 x6 (CD_proj1 x0 x1 x4)) = x2 x3 x4 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
Apply CD_proj0_F with
x0,
x1,
x4,
λ x5 x6 . pair_tag x0 (x2 x3 x6) (x2 0 (CD_proj1 x0 x1 x4)) = x2 x3 x4 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H4.
Apply CD_proj1_F with
x0,
x1,
x4,
λ x5 x6 . pair_tag x0 (x2 x3 x4) (x2 0 x6) = x2 x3 x4 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H4.
Apply H2 with
λ x5 x6 . pair_tag x0 (x2 x3 x4) x6 = x2 x3 x4.
Apply pair_tag_0 with
x0,
x1,
x2 x3 x4.
The subproof is completed by applying H0.