Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι be given.
Apply explicit_Nats_E with
x0,
x1,
x2,
∀ x3 : ι → ο . ∀ x4 . x3 x4 ⟶ ∀ x5 : ι → ι → ι . (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x3 x7 ⟶ x3 (x5 x6 x7)) ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 (explicit_Nats_primrec x0 x1 x2 x4 x5 x6).
Assume H1: x1 ∈ x0.
Assume H2: ∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x0.
Assume H3: ∀ x3 . x3 ∈ x0 ⟶ x2 x3 = x1 ⟶ ∀ x4 : ο . x4.
Assume H4: ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4.
Assume H5: ∀ x3 : ι → ο . x3 x1 ⟶ (∀ x4 . x3 x4 ⟶ x3 (x2 x4)) ⟶ ∀ x4 . x4 ∈ x0 ⟶ x3 x4.
Let x3 of type ι → ο be given.
Let x4 of type ι be given.
Assume H6: x3 x4.
Let x5 of type ι → ι → ι be given.
Assume H7: ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x3 x7 ⟶ x3 (x5 x6 x7).
Apply explicit_Nats_ind with
x0,
x1,
x2,
λ x6 . x3 (explicit_Nats_primrec x0 x1 x2 x4 x5 x6) leaving 3 subgoals.
The subproof is completed by applying H0.
Apply explicit_Nats_primrec_base with
x0,
x1,
x2,
x4,
x5,
λ x6 x7 . x3 x7 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H6.
Let x6 of type ι be given.
Assume H8: x6 ∈ x0.
Apply explicit_Nats_primrec_S with
x0,
x1,
x2,
x4,
x5,
x6,
λ x7 x8 . x3 x8 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H8.
Apply H7 with
x6,
explicit_Nats_primrec x0 x1 x2 x4 x5 x6 leaving 2 subgoals.
The subproof is completed by applying H8.
The subproof is completed by applying H9.