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: ∀ x6 . x1 x6 ⟶ x1 (x2 x6).
Assume H2: ∀ x6 . x1 x6 ⟶ x1 (x3 x6).
Assume H3: ∀ x6 x7 . x1 x6 ⟶ x1 x7 ⟶ x1 (x4 x6 x7).
Assume H4: ∀ x6 x7 . x1 x6 ⟶ x1 x7 ⟶ x1 (x5 x6 x7).
Let x6 of type ι be given.
Let x7 of type ι be given.
Claim L14: x1 (x5 ... ...)
Apply H1 with
x5 (x3 (CD_proj1 x0 x1 x7)) (CD_proj1 x0 x1 x6).
The subproof is completed by applying L14.
Apply H3 with
x5 (CD_proj0 x0 x1 x6) (CD_proj0 x0 x1 x7),
x2 (x5 (x3 (CD_proj1 x0 x1 x7)) (CD_proj1 x0 x1 x6)) leaving 2 subgoals.
The subproof is completed by applying L13.
The subproof is completed by applying L15.
Apply H4 with
CD_proj1 x0 x1 x7,
CD_proj0 x0 x1 x6 leaving 2 subgoals.
The subproof is completed by applying L10.
The subproof is completed by applying L7.
Apply H4 with
CD_proj1 x0 x1 x6,
x3 (CD_proj0 x0 x1 x7) leaving 2 subgoals.
The subproof is completed by applying L8.
The subproof is completed by applying L11.
Apply H3 with
x5 (CD_proj1 x0 x1 x7) (CD_proj0 x0 x1 x6),
x5 (CD_proj1 x0 x1 x6) (x3 (CD_proj0 x0 x1 x7)) leaving 2 subgoals.
The subproof is completed by applying L17.
The subproof is completed by applying L18.
Apply CD_proj0_2 with
x0,
x1,
x4 (x5 (CD_proj0 x0 x1 x6) (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) (CD_proj0 x0 x1 x6)) (x5 (CD_proj1 x0 x1 x6) (x3 (CD_proj0 x0 x1 x7))) leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L16.
The subproof is completed by applying L19.