Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι → ι be given.
Let x3 of type ι → ι → ι be given.
Assume H1: x0 ⊆ x1.
Assume H2: ∀ x4 . x4 ∈ x1 ⟶ ∀ x5 . x5 ∈ x1 ⟶ x2 x4 x5 = x3 x4 x5.
Apply explicit_Group_repindep_imp with
x1,
x2,
x3 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H0.
Apply prop_ext_2 with
explicit_normal x1 x2 x0,
explicit_normal x1 x3 x0 leaving 2 subgoals.
Apply explicit_normal_repindep_imp with
x0,
x1,
x2,
x3 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply explicit_normal_repindep_imp with
x0,
x1,
x3,
x2 leaving 3 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying H1.
Let x4 of type ι be given.
Assume H4: x4 ∈ x1.
Let x5 of type ι be given.
Assume H5: x5 ∈ x1.
Let x6 of type ι → ι → ο be given.
Apply H2 with
x4,
x5,
λ x7 x8 . x6 x8 x7 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H5.