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 → ι → ι be given.
Apply explicit_Nats_E with
x0,
x1,
x2,
∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . c3e2e.. x0 x1 x2 x3 x4 (x2 x5) x6 ⟶ ∃ x7 . and (x6 = x4 x5 x7) (c3e2e.. x0 x1 x2 x3 x4 x5 x7).
Assume H1: x1 ∈ x0.
Assume H2: ∀ x5 . x5 ∈ x0 ⟶ x2 x5 ∈ x0.
Assume H3: ∀ x5 . x5 ∈ x0 ⟶ x2 x5 = x1 ⟶ ∀ x6 : ο . x6.
Assume H4: ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x2 x5 = x2 x6 ⟶ x5 = x6.
Assume H5: ∀ x5 : ι → ο . x5 x1 ⟶ (∀ x6 . x5 x6 ⟶ x5 (x2 x6)) ⟶ ∀ x6 . x6 ∈ x0 ⟶ x5 x6.
Claim L7:
∀ x5 . ... ⟶ ∀ x6 . ... ⟶ (λ x7 x8 . and (c3e2e.. x0 x1 x2 x3 x4 x7 x8) (∀ x9 . ... ⟶ x7 = ... ⟶ ∃ x10 . and (x8 = x4 x9 x10) (c3e2e.. x0 x1 x2 x3 x4 x9 x10))) ... ...Let x5 of type ι be given.
Assume H8: x5 ∈ x0.
Let x6 of type ι be given.
Assume H9:
c3e2e.. x0 x1 x2 x3 x4 (x2 x5) x6.
Claim L10:
∀ x7 . x7 ∈ x0 ⟶ x2 x5 = x2 x7 ⟶ ∃ x8 . and (x6 = x4 x7 x8) (c3e2e.. x0 x1 x2 x3 x4 x7 x8)Apply H9 with
λ x7 x8 . and (c3e2e.. x0 x1 x2 x3 x4 x7 x8) (∀ x9 . x9 ∈ x0 ⟶ x7 = x2 x9 ⟶ ∃ x10 . and (x8 = x4 x9 x10) (c3e2e.. x0 x1 x2 x3 x4 x9 x10)),
∀ x7 . x7 ∈ x0 ⟶ x2 x5 = x2 x7 ⟶ ∃ x8 . and (x6 = x4 x7 x8) (c3e2e.. x0 x1 x2 x3 x4 x7 x8) leaving 3 subgoals.
The subproof is completed by applying L6.
The subproof is completed by applying L7.
Assume H10:
c3e2e.. x0 x1 x2 x3 x4 (x2 x5) x6.
Assume H11:
∀ x7 . x7 ∈ x0 ⟶ x2 x5 = x2 x7 ⟶ ∃ x8 . and (x6 = x4 x7 x8) (c3e2e.. x0 x1 x2 x3 x4 x7 x8).
The subproof is completed by applying H11.
Apply L10 with
x5 leaving 2 subgoals.
The subproof is completed by applying H8.
Let x7 of type ι → ι → ο be given.
Assume H11: x7 (x2 x5) (x2 x5).
The subproof is completed by applying H11.