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_CRing_with_id_E with x0, x1, x2, x3, x4, explicit_CRing_with_id x0 x1 x2 x5 x6.

Assume H2: explicit_CRing_with_id x0 x1 x2 x3 x4.

Assume H3: ∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ prim1 (x3 x7 x8) x0.

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 H6: prim1 x1 x0.

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 H9: ∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ prim1 (x4 x7 x8) x0.

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 H11: ∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x4 x7 x8 = x4 x8 x7.

Assume H12: prim1 x2 x0.

Assume H13: x2 = x1 ⟶ ∀ x7 : ο . x7.

Assume H14: ∀ x7 . prim1 x7 x0 ⟶ x4 x2 x7 = 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).

Apply explicit_CRing_with_id_I with x0, x1, x2, x5, x6 leaving 13 subgoals.

Let x7 of type ι be given.

Assume H16: prim1 x7 x0.

Let x8 of type ι be given.

Assume H17: prim1 x8 x0.

Apply H0 with x7, x8, λ x9 x10 . prim1 x9 x0 leaving 3 subgoals.

The subproof is completed by applying H16.

The subproof is completed by applying H17.

Apply H3 with x7, x8 leaving 2 subgoals.