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 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ explicit_Nats_one_plus x0 x1 x2 x3 x4 ∈ x0.
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 be given.
Assume H6: x3 ∈ x0.
Let x4 of type ι be given.
Assume H7: x4 ∈ x0.
Apply explicit_Nats_primrec_P with
x0,
x1,
x2,
λ x5 . x5 ∈ x0,
x2 x4,
λ x5 x6 . x2 x6,
x3 leaving 4 subgoals.
The subproof is completed by applying H0.
Apply H2 with
x4.
The subproof is completed by applying H7.
Let x5 of type ι be given.
Assume H8: x5 ∈ x0.
Let x6 of type ι be given.
Assume H9: (λ x7 . x7 ∈ x0) x6.
Apply H2 with
x6.
The subproof is completed by applying H9.
The subproof is completed by applying H6.