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.
Apply explicit_CRing_with_id_E with
x0,
x1,
x2,
x3,
x4,
explicit_Ring_with_id x0 x1 x2 x3 x4.
Assume H2:
∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x3 x5 (x3 x6 x7) = x3 (x3 x5 x6) x7.
Assume H3:
∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x3 x5 x6 = x3 x6 x5.
Assume H5:
∀ x5 . prim1 x5 x0 ⟶ x3 x1 x5 = x5.
Assume H6:
∀ x5 . prim1 x5 x0 ⟶ ∃ x6 . and (prim1 x6 x0) (x3 x5 x6 = x1).
Assume H8:
∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x4 x5 (x4 x6 x7) = x4 (x4 x5 x6) x7.
Assume H9:
∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x4 x5 x6 = x4 x6 x5.
Assume H11: x2 = x1 ⟶ ∀ x5 : ο . x5.
Assume H12:
∀ x5 . prim1 x5 x0 ⟶ x4 x2 x5 = x5.
Assume H13:
∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x4 x5 (x3 x6 x7) = x3 (x4 x5 x6) (x4 x5 x7).
Apply explicit_Ring_with_id_I with
x0,
x1,
x2,
x3,
x4 leaving 14 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
The subproof is completed by applying H10.
The subproof is completed by applying H11.
The subproof is completed by applying H12.
Let x5 of type ι be given.
Apply H9 with
x5,
x2,
λ x6 x7 . x7 = x5 leaving 3 subgoals.
The subproof is completed by applying H14.
The subproof is completed by applying H10.
Apply H12 with
x5.
The subproof is completed by applying H14.
The subproof is completed by applying H13.
Let x5 of type ι be given.
Let x6 of type ι be given.
Let x7 of type ι be given.
set y8 to be ...
set y9 to be ...
Claim L17: ∀ x10 : ι → ο . x10 y9 ⟶ x10 y8
Let x10 of type ι → ο be given.
Assume H17: x10 (x5 (x6 x7 y9) (x6 y8 y9)).
Apply H9 with
...,
...,
... leaving 3 subgoals.
Let x10 of type ι → ι → ο be given.
Apply L17 with
λ x11 . x10 x11 y9 ⟶ x10 y9 x11.
Assume H18: x10 y9 y9.
The subproof is completed by applying H18.