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 . x3 x1 ⟶ (∀ x4 . x4 ∈ x0 ⟶ x3 x4 ⟶ x3 (x2 x4)) ⟶ ∀ x4 . x4 ∈ x0 ⟶ x3 x4.
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.
Assume H6: x3 x1.
Assume H7: ∀ x4 . x4 ∈ x0 ⟶ x3 x4 ⟶ x3 (x2 x4).
Claim L8:
∀ x4 . x4 ∈ x0 ⟶ and (x4 ∈ x0) (x3 x4)Apply H5 with
λ x4 . and (x4 ∈ x0) (x3 x4) leaving 2 subgoals.
Apply andI with
x1 ∈ x0,
x3 x1 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H6.
Let x4 of type ι be given.
Assume H8:
and (x4 ∈ x0) (x3 x4).
Apply H8 with
and (x2 x4 ∈ x0) (x3 (x2 x4)).
Assume H9: x4 ∈ x0.
Assume H10: x3 x4.
Apply andI with
x2 x4 ∈ x0,
x3 (x2 x4) leaving 2 subgoals.
Apply H2 with
x4.
The subproof is completed by applying H9.
Apply H7 with
x4 leaving 2 subgoals.
The subproof is completed by applying H9.
The subproof is completed by applying H10.
Let x4 of type ι be given.
Assume H9: x4 ∈ x0.
Apply L8 with
x4,
x3 x4 leaving 2 subgoals.
The subproof is completed by applying H9.
Assume H10: x4 ∈ x0.
Assume H11: x3 x4.
The subproof is completed by applying H11.