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: ∀ x3 x4 . x1 x3 ⟶ x1 x4 ⟶ x1 (x2 x3 x4).
Let x3 of type ι be given.
Let x4 of type ι be given.
Apply CD_carr_I with
x0,
x1,
x2 (CD_proj0 x0 x1 x3) (CD_proj0 x0 x1 x4),
x2 (CD_proj1 x0 x1 x3) (CD_proj1 x0 x1 x4) leaving 3 subgoals.
The subproof is completed by applying H0.
Apply H1 with
CD_proj0 x0 x1 x3,
CD_proj0 x0 x1 x4 leaving 2 subgoals.
Apply CD_proj0R with
x0,
x1,
x3 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Apply CD_proj0R with
x0,
x1,
x4 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H3.
Apply H1 with
CD_proj1 x0 x1 x3,
CD_proj1 x0 x1 x4 leaving 2 subgoals.
Apply CD_proj1R with
x0,
x1,
x3 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Apply CD_proj1R with
x0,
x1,
x4 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H3.