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 H1: ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 ∈ x0.
Assume H2: ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x5 (x3 x6 x7) = x3 (x3 x5 x6) x7.
Assume H3: ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 = x3 x6 x5.
Assume H4: x1 ∈ x0.
Assume H5: ∀ x5 . x5 ∈ x0 ⟶ x3 x1 x5 = x5.
Assume H6:
∀ x5 . x5 ∈ x0 ⟶ ∃ x6 . and (x6 ∈ x0) (x3 x5 x6 = x1).
Assume H7: ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 x6 ∈ x0.
Assume H8: ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x5 (x4 x6 x7) = x4 (x4 x5 x6) x7.
Assume H9: ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 x6 = x4 x6 x5.
Assume H10: x2 ∈ x0.
Assume H11: x2 = x1 ⟶ ∀ x5 : ο . x5.
Assume H12: ∀ x5 . x5 ∈ x0 ⟶ x4 x2 x5 = x5.
Assume H13: ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . 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.
Assume H14: x5 ∈ x0.
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.
Assume H14: x5 ∈ x0.
Let x6 of type ι be given.
Assume H15: x6 ∈ x0.
Let x7 of type ι be given.
Assume H16: x7 ∈ x0.
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
x5 x7 y8,
y9,
λ x11 . x10 leaving 3 subgoals.
Apply H1 with
x7,
y8 leaving 2 subgoals.
The subproof is completed by applying H14.
The subproof is completed by applying H15.
The subproof is completed by applying H16.
Apply H13 with
y9,
x7,
y8,
λ x11 . x10 leaving 4 subgoals.
The subproof is completed by applying H16.
The subproof is completed by applying H14.
The subproof is completed by applying H15.
set y11 to be ...
set y12 to be ...
set y13 to be ...
Apply L18 with
λ x14 . ... ⟶ y13 ... ... leaving 2 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.