Apply andI with
∀ x0 . x0 ∈ 1 ⟶ ∀ x1 . x1 ∈ 1 ⟶ ∀ x2 . x2 ∈ 1 ⟶ 0 = 0,
∃ x0 . and (x0 ∈ 1) (∀ x1 . x1 ∈ 1 ⟶ and (0 = x1) (0 = x1)) leaving 2 subgoals.
Let x0 of type ι be given.
Assume H1: x0 ∈ 1.
Let x1 of type ι be given.
Assume H2: x1 ∈ 1.
Let x2 of type ι be given.
Assume H3: x2 ∈ 1.
Let x3 of type ι → ι → ο be given.
Assume H4: x3 0 0.
The subproof is completed by applying H4.
Let x0 of type ο be given.
Assume H1:
∀ x1 . and (x1 ∈ 1) (∀ x2 . ... ⟶ and (0 = ...) ...) ⟶ x0.
Claim L2:
∀ x0 : ι → ι → ι . (∀ x1 . x1 ∈ 1 ⟶ ∀ x2 . x2 ∈ 1 ⟶ 0 = x0 x1 x2) ⟶ 0941b.. 1 x0Let x0 of type ι → ι → ι be given.
Assume H2: ∀ x1 . x1 ∈ 1 ⟶ ∀ x2 . x2 ∈ 1 ⟶ 0 = x0 x1 x2.
Apply andI with
∀ x1 . x1 ∈ 1 ⟶ ∀ x2 . x2 ∈ 1 ⟶ ∀ x3 . x3 ∈ 1 ⟶ x0 (x0 x1 x2) x3 = x0 x1 (x0 x2 x3),
∃ x1 . and (x1 ∈ 1) (∀ x2 . x2 ∈ 1 ⟶ and (x0 x2 x1 = x2) (x0 x1 x2 = x2)) leaving 2 subgoals.
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 H2 with
x1,
x2,
λ x4 x5 . x0 x4 x3 = x0 x1 (x0 x2 x3) leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
Apply H2 with
x2,
x3,
λ x4 x5 . x0 0 x3 = x0 x1 x4 leaving 3 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
Apply H2 with
0,
x3,
λ x4 x5 . x4 = x0 x1 0 leaving 3 subgoals.
The subproof is completed by applying In_0_1.
The subproof is completed by applying H5.
Apply H2 with
x1,
0 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying In_0_1.
Let x1 of type ο be given.
Assume H3:
∀ x2 . and (x2 ∈ 1) (∀ x3 . x3 ∈ 1 ⟶ and (x0 x3 x2 = x3) (x0 x2 x3 = x3)) ⟶ x1.
Apply H3 with
0.
Apply andI with
0 ∈ 1,
∀ x2 . x2 ∈ 1 ⟶ and (x0 x2 0 = x2) (x0 0 x2 = x2) leaving 2 subgoals.
The subproof is completed by applying In_0_1.
Let x2 of type ι be given.
Assume H4: x2 ∈ 1.
Apply cases_1 with
x2,
λ x3 . and (x0 x3 0 = x3) (x0 0 x3 = x3) leaving 2 subgoals.
The subproof is completed by applying H4.
Apply andI with
x0 0 0 = 0,
x0 0 0 = 0 leaving 2 subgoals.
Let x3 of type ι → ι → ο be given.
Apply H2 with
0,
0,
λ x4 x5 . x3 x5 x4 leaving 2 subgoals.
The subproof is completed by applying In_0_1.
The subproof is completed by applying In_0_1.
Let x3 of type ι → ι → ο be given.
Apply H2 with
0,
0,
λ x4 x5 . x3 x5 x4 leaving 2 subgoals.
The subproof is completed by applying In_0_1.
The subproof is completed by applying In_0_1.
Apply unknownprop_6482bf17c7629de0c611c16b71ae30c036294b46cbc3e9f673f7271f20ce0d70 with
0941b.. leaving 2 subgoals.
The subproof is completed by applying L0.
The subproof is completed by applying L2.