Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι be given.
Let x3 of type ι be given.
Let x4 of type ι be given.
Let x5 of type ι → ι be given.
Let x6 of type ι → ι be given.
Apply explicit_Nats_E with
x0,
x1,
x2,
bij x0 x3 x6 ⟶ x6 x1 = x4 ⟶ (∀ x7 . x7 ∈ x0 ⟶ x6 (x2 x7) = x5 (x6 x7)) ⟶ explicit_Nats x3 x4 x5.
Assume H1: x1 ∈ x0.
Assume H2: ∀ x7 . x7 ∈ x0 ⟶ x2 x7 ∈ x0.
Assume H3: ∀ x7 . x7 ∈ x0 ⟶ x2 x7 = x1 ⟶ ∀ x8 : ο . x8.
Assume H4: ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x2 x7 = x2 x8 ⟶ x7 = x8.
Assume H5: ∀ x7 : ι → ο . x7 x1 ⟶ (∀ x8 . x7 x8 ⟶ x7 (x2 x8)) ⟶ ∀ x8 . x8 ∈ x0 ⟶ x7 x8.
Assume H7: x6 x1 = x4.
Assume H8: ∀ x7 . x7 ∈ x0 ⟶ x6 (x2 x7) = x5 (x6 x7).
Apply H6 with
explicit_Nats x3 x4 x5.
Assume H9:
and (∀ x7 . x7 ∈ x0 ⟶ x6 x7 ∈ x3) (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x6 x7 = x6 x8 ⟶ x7 = x8).
Apply H9 with
(∀ x7 . x7 ∈ x3 ⟶ ∃ x8 . and (x8 ∈ x0) (x6 x8 = x7)) ⟶ explicit_Nats x3 x4 x5.
Assume H10: ∀ x7 . x7 ∈ x0 ⟶ x6 x7 ∈ x3.
Assume H11: ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x6 x7 = x6 x8 ⟶ x7 = x8.
Assume H12:
∀ x7 . x7 ∈ x3 ⟶ ∃ x8 . and (x8 ∈ x0) (x6 x8 = x7).
Apply explicit_Nats_I with
x3,
x4,
x5 leaving 5 subgoals.
Apply H7 with
λ x7 x8 . x7 ∈ x3.
Apply H10 with
x1.
The subproof is completed by applying H1.
Let x7 of type ι be given.
Assume H13: x7 ∈ x3.
Apply H12 with
x7,
x5 x7 ∈ x3 leaving 2 subgoals.
The subproof is completed by applying H13.
Let x8 of type ι be given.
Assume H14:
(λ x9 . and (x9 ∈ x0) (x6 x9 = x7)) x8.
Apply H14 with
x5 x7 ∈ x3.
Assume H15: x8 ∈ x0.
Assume H16: x6 x8 = x7.
Apply H16 with
λ x9 x10 . x5 x9 ∈ x3.
Apply H8 with
x8,
λ x9 x10 . x9 ∈ x3 leaving 2 subgoals.
The subproof is completed by applying H15.
Apply H10 with
x2 x8.
Apply H2 with
x8.
The subproof is completed by applying H15.
Let x7 of type ι be given.
Assume H13: x7 ∈ x3.
Apply H12 with
x7,
x5 x7 = x4 ⟶ ∀ x8 : ο . x8 leaving 2 subgoals.
The subproof is completed by applying H13.
Let x8 of type ι be given.
Assume H14:
(λ x9 . and (x9 ∈ x0) (x6 x9 = x7)) x8.
Apply H14 with
x5 ... = ... ⟶ ∀ x9 : ο . x9.