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).
Assume H5: x1 0.
Assume H6: x1 1.
Let x6 of type ι be given.
Apply nat_ind with
λ x7 . CD_carr x0 x1 (CD_exp_nat x0 x1 x2 x3 x4 x5 x6 x7) leaving 2 subgoals.
Apply CD_exp_nat_0 with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
λ x7 x8 . CD_carr x0 x1 x8 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply CD_carr_0ext with
x0,
x1,
1 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
Let x7 of type ι be given.
Apply CD_exp_nat_S with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
λ x8 x9 . CD_carr x0 x1 x9 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H8.
Apply CD_mul_CD with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
CD_exp_nat x0 x1 x2 x3 x4 x5 x6 x7 leaving 7 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 H7.
The subproof is completed by applying H9.