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 iffI with
explicit_Group x0 x1,
explicit_Group x0 x2 leaving 2 subgoals.
Apply explicit_Group_repindep_imp with
x0,
x1,
x2.
The subproof is completed by applying H0.
Apply explicit_Group_repindep_imp with
x0,
x2,
x1.
Let x3 of type ι be given.
Assume H1: x3 ∈ x0.
Let x4 of type ι be given.
Assume H2: x4 ∈ x0.
Let x5 of type ι → ι → ο be given.
Apply H0 with
x3,
x4,
λ x6 x7 . x5 x7 x6 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.