Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι → ι → ι be given.
Let x4 of type ι → ι → ι be given.
Let x5 of type ι → ι → ι be given.
Let x6 of type ι → ι → ι be given.
Assume H0:
∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x3 x7 x8 = x5 x7 x8.
Assume H1:
∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x4 x7 x8 = x6 x7 x8.
Apply explicit_Ring_with_id_E with
x0,
x1,
x2,
x3,
x4,
explicit_Ring_with_id x0 x1 x2 x5 x6.
Assume H4:
∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x3 x7 (x3 x8 x9) = x3 (x3 x7 x8) x9.
Assume H5:
∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x3 x7 x8 = x3 x8 x7.
Assume H7:
∀ x7 . prim1 x7 x0 ⟶ x3 x1 x7 = x7.
Assume H8:
∀ x7 . prim1 x7 x0 ⟶ ∃ x8 . and (prim1 x8 x0) (x3 x7 x8 = x1).
Assume H10:
∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x4 x7 (x4 x8 x9) = x4 (x4 x7 x8) x9.
Assume H12: x2 = x1 ⟶ ∀ x7 : ο . x7.
Assume H13:
∀ x7 . prim1 x7 x0 ⟶ x4 x2 x7 = x7.
Assume H14:
∀ x7 . prim1 x7 x0 ⟶ x4 x7 x2 = x7.
Assume H15:
∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x4 x7 (x3 x8 x9) = x3 (x4 x7 x8) (x4 x7 x9).
Assume H16:
∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x4 (x3 x7 x8) x9 = x3 (x4 x7 x9) (x4 x8 x9).
Apply explicit_Ring_with_id_I with
x0,
x1,
x2,
x5,
x6 leaving 14 subgoals.
Let x7 of type ι be given.
Let x8 of type ι be given.