Let x0 of type ι → ι → ι be given.
Assume H0: x0 0 0 = 0.
Apply and3I with
∀ x1 . x1 ∈ 1 ⟶ ∀ x2 . x2 ∈ 1 ⟶ x0 x1 x2 ∈ 1,
∀ x1 . x1 ∈ 1 ⟶ ∀ x2 . x2 ∈ 1 ⟶ ∀ x3 . x3 ∈ 1 ⟶ x0 x1 (x0 x2 x3) = x0 (x0 x1 x2) x3,
∃ x1 . and (x1 ∈ 1) (and (∀ x2 . x2 ∈ 1 ⟶ and (x0 x1 x2 = x2) (x0 x2 x1 = x2)) (∀ x2 . x2 ∈ 1 ⟶ ∃ x3 . and (x3 ∈ 1) (and (x0 x2 x3 = x1) (x0 x3 x2 = x1)))) leaving 3 subgoals.
Let x1 of type ι be given.
Assume H3: x1 ∈ 1.
Let x2 of type ι be given.
Assume H4: x2 ∈ 1.
Apply L2 with
x1,
x2,
λ x3 x4 . x4 ∈ 1 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying In_0_1.
Let x1 of type ι be given.
Assume H3: x1 ∈ 1.
Let x2 of type ι be given.
Assume H4: x2 ∈ 1.
Let x3 of type ι be given.
Assume H5: x3 ∈ 1.
Apply L2 with
x1,
x2,
λ x4 x5 . x0 x1 (x0 x2 x3) = x0 x5 x3 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
Apply L2 with
x2,
x3,
λ x4 x5 . x0 x1 x5 = x0 0 x3 leaving 3 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Apply L2 with
0,
x3,
λ x4 x5 . x0 x1 0 = x5 leaving 3 subgoals.
The subproof is completed by applying In_0_1.
The subproof is completed by applying H5.
Apply L1 with
x1.
The subproof is completed by applying H3.
Let x1 of type ο be given.
Assume H3:
∀ x2 . and (x2 ∈ 1) (and (∀ x3 . x3 ∈ 1 ⟶ and (x0 x2 x3 = x3) (x0 x3 x2 = x3)) (∀ x3 . x3 ∈ 1 ⟶ ∃ x4 . and (x4 ∈ 1) (and (x0 x3 x4 = x2) (x0 x4 x3 = x2)))) ⟶ x1.
Apply H3 with
0.
Apply andI with
0 ∈ 1,
and (∀ x2 . x2 ∈ 1 ⟶ and (x0 0 x2 = x2) (x0 x2 0 = x2)) (∀ x2 . x2 ∈ 1 ⟶ ∃ x3 . and (x3 ∈ 1) (and (x0 x2 x3 = 0) (x0 x3 x2 = 0))) leaving 2 subgoals.
The subproof is completed by applying In_0_1.
Apply andI with
∀ x2 . x2 ∈ 1 ⟶ and (x0 0 x2 = x2) (x0 x2 0 = x2),
∀ x2 . x2 ∈ 1 ⟶ ∃ x3 . and (x3 ∈ 1) (and (x0 x2 x3 = 0) (x0 x3 x2 = 0)) leaving 2 subgoals.
Let x2 of type ι be given.
Assume H4: x2 ∈ 1.
Apply cases_1 with
x2,
λ x3 . and (x0 0 x3 = x3) (x0 ... 0 = ...) leaving 2 subgoals.