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.
Let x4 of type ο be given.
Assume H0:
explicit_Ring x0 x1 x2 x3 ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x2 x5 x6 ∈ x0) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x2 x5 (x2 x6 x7) = x2 (x2 x5 x6) x7) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x2 x5 x6 = x2 x6 x5) ⟶ x1 ∈ x0 ⟶ (∀ x5 . x5 ∈ x0 ⟶ x2 x1 x5 = x5) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∃ x6 . and (x6 ∈ x0) (x2 x5 x6 = x1)) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 ∈ x0) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x5 (x3 x6 x7) = x3 (x3 x5 x6) x7) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x5 (x2 x6 x7) = x2 (x3 x5 x6) (x3 x5 x7)) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 (x2 x5 x6) x7 = x2 (x3 x5 x7) (x3 x6 x7)) ⟶ x4.
Apply and4E with
and (and (and (and (and (and (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x2 x5 x6 ∈ x0) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x2 x5 (x2 x6 x7) = x2 (x2 x5 x6) x7)) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x2 x5 x6 = x2 x6 x5)) (x1 ∈ x0)) (∀ x5 . x5 ∈ x0 ⟶ x2 x1 x5 = x5)) (∀ x5 . x5 ∈ x0 ⟶ ∃ x6 . and (x6 ∈ x0) (x2 x5 x6 = x1))) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 ∈ x0),
∀ x5 . ... ⟶ ∀ x6 . ... ⟶ ∀ x7 . ...,
...,
...,
... leaving 2 subgoals.