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_Field_E with
x0,
x1,
x2,
x3,
x4,
explicit_CRing_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 ⟶ (x5 = x1 ⟶ ∀ x6 : ο . x6) ⟶ ∃ x6 . and (x6 ∈ x0) (x4 x5 x6 = x2).
Assume H14: ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x5 (x3 x6 x7) = x3 (x4 x5 x6) (x4 x5 x7).
Apply explicit_CRing_with_id_I with
x0,
x1,
x2,
x3,
x4 leaving 13 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 H9.
The subproof is completed by applying H10.
The subproof is completed by applying H11.
The subproof is completed by applying H12.
The subproof is completed by applying H14.