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.

Assume H2: CD_carr x0 x1 x3.

Assume H3: CD_carr x0 x1 x4.

Claim L4: x1 (CD_proj0 x0 x1 x3)

Apply CD_proj0R with x0, x1, x3 leaving 2 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H2.

Claim L5: x1 (CD_proj1 x0 x1 x3)

Apply CD_proj1R with x0, x1, x3 leaving 2 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H2.

Claim L6: x1 (CD_proj0 x0 x1 x4)

Apply CD_proj0R with x0, x1, x4 leaving 2 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H3.

Claim L7: x1 (CD_proj1 x0 x1 x4)

Apply CD_proj1R with x0, x1, x4 leaving 2 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H3.

Claim L8: x1 (x2 (CD_proj0 x0 x1 x3) (CD_proj0 x0 x1 x4))

Apply H1 with CD_proj0 x0 x1 x3, CD_proj0 x0 x1 x4 leaving 2 subgoals.

The subproof is completed by applying L4.

The subproof is completed by applying L6.

Claim L9: x1 (x2 (CD_proj1 x0 x1 x3) (CD_proj1 x0 x1 x4))

Apply H1 with CD_proj1 x0 x1 x3, CD_proj1 x0 x1 x4 leaving 2 subgoals.

The subproof is completed by applying L5.

The subproof is completed by applying L7.

Apply CD_proj1_2 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.

The subproof is completed by applying L8.

The subproof is completed by applying L9.

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Proofgold Proof