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,
∀ x5 . prim1 x5 x0 ⟶ x4 x5 x5 = x1 ⟶ x5 = x1.
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 ⟶ (x5 = x1 ⟶ ∀ x6 : ο . x6) ⟶ ∃ x6 . and (prim1 x6 x0) (x4 x5 x6 = x2).
Assume H14:
∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x4 x5 (x3 x6 x7) = x3 (x4 x5 x6) (x4 x5 x7).
Let x5 of type ι be given.
Assume H16: x4 x5 x5 = x1.
Apply dneg with
x5 = x1.
Assume H17: x5 = x1 ⟶ ∀ x6 : ο . x6.
Apply H13 with
x5,
False leaving 3 subgoals.
The subproof is completed by applying H15.
The subproof is completed by applying H17.
Let x6 of type ι be given.
Assume H18:
(λ x7 . and (prim1 x7 x0) (x4 x5 x7 = x2)) x6.
Apply H18 with
False.
Assume H20: x4 ... ... = ....