Claim L0:
∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ∈ x0) ⟶ ∀ x2 : ι → ι → ι . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 = x2 x3 x4) ⟶ 127dd.. x0 x2 = 127dd.. x0 x1Let x0 of type ι be given.
Let x1 of type ι → ι → ι be given.
Assume H0: ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ∈ x0.
The subproof is completed by applying unknownprop_f057e15a1f9fd74889f75f27da9c2bb26d9ceda8f183ecded8c416471b144fea with x0, x1.
Claim L1:
∀ x0 : ι → ι → ι . (∀ x1 . x1 ∈ 1 ⟶ ∀ x2 . x2 ∈ 1 ⟶ 0 = x0 x1 x2) ⟶ 127dd.. 1 x0Let x0 of type ι → ι → ι be given.
Assume H1: ∀ x1 . x1 ∈ 1 ⟶ ∀ x2 . x2 ∈ 1 ⟶ 0 = x0 x1 x2.
Apply andI with
explicit_Group 1 x0,
explicit_abelian 1 x0 leaving 2 subgoals.
Apply unknownprop_5afb90ce2dcd7d6069f28fcfc6fd878930e4971602c481fb6b43316654d0aad8 with
x0.
Let x1 of type ι → ι → ο be given.
Apply H1 with
0,
0,
λ x2 x3 . x1 x3 x2 leaving 2 subgoals.
The subproof is completed by applying In_0_1.
The subproof is completed by applying In_0_1.
Let x1 of type ι be given.
Assume H2: x1 ∈ 1.
Let x2 of type ι be given.
Assume H3: x2 ∈ 1.
Apply H1 with
x1,
x2,
λ x3 x4 . x3 = x0 x2 x1 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Apply H1 with
x2,
x1 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H2.
Apply unknownprop_6482bf17c7629de0c611c16b71ae30c036294b46cbc3e9f673f7271f20ce0d70 with
127dd.. leaving 2 subgoals.
The subproof is completed by applying L0.
The subproof is completed by applying L1.