Let x0 of type ι be given.
Let x1 of type ι → ι → ι be given.
Let x2 of type ι → ι → ι be given.
Assume H0: ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 = x2 x3 x4.
Apply explicit_Group_repindep_imp with
x0,
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply explicit_Group_identity_in with
x0,
x1.
The subproof is completed by applying H1.
Apply explicit_Group_identity_lid with
x0,
x1.
The subproof is completed by applying H1.
Apply explicit_Group_identity_in with
x0,
x2.
The subproof is completed by applying L2.
Apply explicit_Group_identity_rid with
x0,
x2.
The subproof is completed by applying L2.
Let x3 of type ι be given.
Assume H7: x3 ∈ x0.
Apply H0 with
x3,
explicit_Group_identity x0 x2,
λ x4 x5 . x5 = x3 leaving 3 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying L5.
Apply L6 with
x3.
The subproof is completed by applying H7.
Apply explicit_Group_identity_unique with
x0,
x1,
explicit_Group_identity x0 x1,
explicit_Group_identity x0 x2 leaving 4 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying L5.
The subproof is completed by applying L4.
The subproof is completed by applying L7.