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.
Let x5 of type ι be given.
Let x6 of type ι be given.
Let x7 of type ι be given.
Let x8 of type ι → ι → ι be given.
Let x9 of type ι → ι → ι be given.
Let x10 of type ι → ι be given.
Apply explicit_Field_E with
x0,
x1,
x2,
x3,
x4,
bij x0 x5 x10 ⟶ x10 x1 = x6 ⟶ x10 x2 = x7 ⟶ (∀ x11 . prim1 x11 x0 ⟶ ∀ x12 . prim1 x12 x0 ⟶ x10 (x3 x11 x12) = x8 (x10 x11) (x10 x12)) ⟶ (∀ x11 . prim1 x11 x0 ⟶ ∀ x12 . prim1 x12 x0 ⟶ x10 (x4 x11 x12) = x9 (x10 x11) (x10 x12)) ⟶ explicit_Field x5 x6 x7 x8 x9.
Assume H1:
∀ x11 . prim1 x11 x0 ⟶ ∀ x12 . prim1 x12 x0 ⟶ prim1 (x3 x11 x12) x0.
Assume H2:
∀ x11 . prim1 x11 x0 ⟶ ∀ x12 . prim1 x12 x0 ⟶ ∀ x13 . prim1 x13 x0 ⟶ x3 x11 (x3 x12 x13) = x3 (x3 x11 x12) x13.
Assume H3:
∀ x11 . prim1 x11 x0 ⟶ ∀ x12 . prim1 x12 x0 ⟶ x3 x11 x12 = x3 x12 x11.
Assume H5:
∀ x11 . prim1 x11 x0 ⟶ x3 x1 x11 = x11.
Assume H6:
∀ x11 . prim1 x11 x0 ⟶ ∃ x12 . and (prim1 x12 x0) (x3 x11 x12 = x1).
Assume H7:
∀ x11 . prim1 x11 x0 ⟶ ∀ x12 . prim1 x12 x0 ⟶ prim1 (x4 x11 x12) x0.
Assume H8:
∀ x11 . prim1 x11 x0 ⟶ ∀ x12 . prim1 x12 x0 ⟶ ∀ x13 . prim1 x13 x0 ⟶ x4 x11 (x4 x12 x13) = x4 (x4 x11 x12) x13.
Assume H9:
∀ x11 . prim1 x11 x0 ⟶ ∀ x12 . prim1 x12 x0 ⟶ x4 x11 x12 = x4 x12 x11.
Assume H11: x2 = x1 ⟶ ∀ x11 : ο . x11.
Assume H12:
∀ x11 . prim1 x11 x0 ⟶ x4 x2 x11 = x11.
Assume H13:
∀ x11 . prim1 x11 x0 ⟶ (x11 = x1 ⟶ ∀ x12 : ο . x12) ⟶ ∃ x12 . and (prim1 x12 x0) (x4 x11 x12 = x2).
Assume H14:
∀ x11 . ... ⟶ ∀ x12 . ... ⟶ ∀ x13 . prim1 x13 x0 ⟶ x4 x11 (x3 x12 x13) = x3 (x4 x11 x12) (x4 x11 x13).