Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι → ι be given.
Let x3 of type ι → ι → ι be given.
Apply explicit_Ring_E with
x0,
x1,
x2,
x3,
∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x4 x5 = x2 x4 x6 ⟶ x5 = x6.
Assume H2:
∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x4 (x2 x5 x6) = x2 (x2 x4 x5) x6.
Assume H3:
∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x2 x4 x5 = x2 x5 x4.
Assume H5:
∀ x4 . prim1 x4 x0 ⟶ x2 x1 x4 = x4.
Assume H6:
∀ x4 . prim1 x4 x0 ⟶ ∃ x5 . and (prim1 x5 x0) (x2 x4 x5 = x1).
Assume H8:
∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x3 x4 (x3 x5 x6) = x3 (x3 x4 x5) x6.
Assume H9:
∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x3 x4 (x2 x5 x6) = x2 (x3 x4 x5) (x3 x4 x6).
Assume H10:
∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x3 (x2 x4 x5) x6 = x2 (x3 x4 x6) (x3 x5 x6).
Let x4 of type ι be given.
Let x5 of type ι be given.
Let x6 of type ι be given.
Assume H14: x2 x4 x5 = x2 x4 x6.
Apply H5 with
x5,
λ x7 x8 . x7 = x6 leaving 2 subgoals.
The subproof is completed by applying H12.
Apply H5 with
x6,
λ x7 x8 . x2 x1 x5 = x7 leaving 2 subgoals.
The subproof is completed by applying H13.
Apply explicit_Ring_minus_L with
x0,
x1,
x2,
x3,
x4,
λ x7 x8 . x2 x7 x5 = x2 x7 x6 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H11.