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.
Let x4 of type ι be given.
Assume H4: x4 ∈ x1.
Let x5 of type ι be given.
Apply ReplE_impred with
x0,
λ x6 . x3 x4 (x3 x6 (explicit_Group_inverse x1 x3 x4)),
x5,
x5 ∈ x0 leaving 2 subgoals.
The subproof is completed by applying H5.
Let x6 of type ι be given.
Assume H6: x6 ∈ x0.
Apply H0 with
x2 x6 (explicit_Group_inverse x1 x2 x4) ∈ x1.
Assume H10:
and (∀ x7 . x7 ∈ x1 ⟶ ∀ x8 . x8 ∈ x1 ⟶ x2 x7 x8 ∈ x1) (∀ x7 . x7 ∈ x1 ⟶ ∀ x8 . x8 ∈ x1 ⟶ ∀ x9 . x9 ∈ x1 ⟶ x2 x7 (x2 x8 x9) = x2 (x2 x7 x8) x9).
Assume H11:
∃ x7 . and (x7 ∈ x1) (and (∀ x8 . ...) ...).
Apply H7 with
λ x7 x8 . x8 = x2 x4 (x2 x6 (explicit_Group_inverse x1 x2 x4)).
Apply L8 with
λ x7 x8 . x3 x4 (x3 x6 x7) = x2 x4 (x2 x6 (explicit_Group_inverse x1 x2 x4)).
Apply H2 with
x6,
explicit_Group_inverse x1 x2 x4,
λ x7 x8 . x3 x4 x7 = x2 x4 (x2 x6 (explicit_Group_inverse x1 x2 x4)) leaving 3 subgoals.
Apply H1 with
x6.
The subproof is completed by applying H6.
The subproof is completed by applying L9.
Apply H2 with
x4,
x2 x6 (explicit_Group_inverse x1 x2 x4),
λ x7 x8 . x7 = x2 x4 (x2 x6 (explicit_Group_inverse x1 x2 x4)) leaving 3 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying L10.
Let x7 of type ι → ι → ο be given.
The subproof is completed by applying H11.
Apply H3 with
x4,
x5 leaving 2 subgoals.
The subproof is completed by applying H4.