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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ιο be given.
Assume H0: ∀ x2 . x1 x2∀ x3 . x3x2nIn x0 x3.
Let x2 of type ιι be given.
Let x3 of type ιι be given.
Let x4 of type ιιι be given.
Let x5 of type ιιι 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 x6x4 0 x6 = x6.
Assume H7: ∀ x6 . x1 x6x4 x6 0 = x6.
Assume H8: ∀ x6 . x1 x6x5 0 x6 = 0.
Assume H9: ∀ x6 . x1 x6x5 x6 1 = x6.
Let x6 of type ι be given.
Assume H10: CD_carr x0 x1 x6.
Apply CD_exp_nat_S with x0, x1, x2, x3, x4, x5, x6, 0, λ x7 x8 . x8 = x6 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying nat_0.
Apply CD_exp_nat_0 with x0, x1, x2, x3, x4, x5, x6, λ x7 x8 . CD_mul x0 x1 x2 x3 x4 x5 x6 x8 = x6 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply CD_mul_1R with x0, x1, x2, x3, x4, x5, x6 leaving 11 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.