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: x1 1.
Assume H3: x2 0 = 0.
Assume H4: x3 0 = 0.
Assume H5: x3 1 = 1.
Assume H6: ∀ x6 . x1 x6 ⟶ x4 0 x6 = x6.
Assume H7: ∀ x6 . x1 x6 ⟶ x4 x6 0 = x6.
Assume H8: ∀ x6 . x1 x6 ⟶ x5 0 x6 = 0.
Assume H9: ∀ x6 . x1 x6 ⟶ x5 x6 1 = x6.
Let x6 of type ι be given.
Apply CD_exp_nat_S with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
1,
λ x7 x8 . x8 = CD_mul x0 x1 x2 x3 x4 x5 x6 x6 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying nat_1.
set y8 to be
CD_mul x1 x2 x3 x4 x5 x6 y7 y7Claim L11: ∀ x9 : ι → ο . x9 y8 ⟶ x9 y7
Let x9 of type ι → ο be given.
Assume H11:
x9 (CD_mul x2 x3 x4 x5 x6 y7 y8 y8).
set y10 to be λ x10 . x9
Apply CD_exp_nat_1 with
x2,
x3,
x4,
x5,
x6,
y7,
y8,
λ x11 x12 . y10 (CD_mul x2 x3 x4 x5 x6 y7 y8 x11) (CD_mul x2 x3 x4 x5 x6 y7 y8 x12) leaving 12 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
The subproof is completed by applying H10.
The subproof is completed by applying H11.
Let x9 of type ι → ι → ο be given.
Apply L11 with
λ x10 . x9 x10 y8 ⟶ x9 y8 x10.
Assume H12: x9 y8 y8.
The subproof is completed by applying H12.