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 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x2 x5 x6) x0) ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x2 x5 (x2 x6 x7) = x2 (x2 x5 x6) x7) ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x5 x6 = x2 x6 x5) ⟶ prim1 x1 x0 ⟶ (∀ x5 . prim1 x5 x0 ⟶ x2 x1 x5 = x5) ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∃ x6 . and (prim1 x6 x0) (x2 x5 x6 = x1)) ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x3 x5 x6) x0) ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x3 x5 (x3 x6 x7) = x3 (x3 x5 x6) x7) ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x3 x5 (x2 x6 x7) = x2 (x3 x5 x6) (x3 x5 x7)) ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 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 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x2 x5 x6) x0) (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x2 x5 (x2 x6 x7) = x2 (x2 x5 x6) x7)) (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x5 x6 = x2 x6 x5)) (prim1 x1 x0)) (∀ x5 . prim1 x5 x0 ⟶ x2 x1 x5 = x5)) (∀ x5 . ... ⟶ ∃ x6 . and (prim1 x6 x0) (x2 x5 x6 = ...))) ...,
...,
...,
...,
... leaving 2 subgoals.