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 → ι → ο be given.
Let x6 of type ι → ι → ι be given.
Let x7 of type ι → ι → ι be given.
Let x8 of type ι → ι → ο be given.
Assume H0: ∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ x3 x9 x10 = x6 x9 x10.
Assume H1: ∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ x4 x9 x10 = x7 x9 x10.
Assume H2:
∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ iff (x5 x9 x10) (x8 x9 x10).
Apply explicit_OrderedField_E with
x0,
x1,
x2,
x3,
x4,
x5,
explicit_OrderedField x0 x1 x2 x6 x7 x8.
Apply explicit_Field_E with
x0,
x1,
x2,
x3,
x4,
(∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ x5 x9 x10 ⟶ x5 x10 x11 ⟶ x5 x9 x11) ⟶ (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ iff (and (x5 x9 x10) (x5 x10 x9)) (x9 = x10)) ⟶ (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ or (x5 x9 x10) (x5 x10 x9)) ⟶ (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ x5 x9 x10 ⟶ x5 (x3 x9 x11) (x3 x10 x11)) ⟶ (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ x5 x1 x9 ⟶ x5 x1 x10 ⟶ x5 x1 (x4 x9 x10)) ⟶ explicit_OrderedField x0 x1 x2 x6 x7 x8.
Assume H5: ∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ x3 x9 x10 ∈ x0.
Assume H6: ∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ x3 x9 (x3 x10 x11) = x3 (x3 x9 x10) x11.
Assume H7: ∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ x3 x9 x10 = x3 x10 x9.
Assume H8: x1 ∈ x0.
Assume H9: ∀ x9 . x9 ∈ x0 ⟶ x3 x1 x9 = x9.
Assume H10:
∀ x9 . x9 ∈ x0 ⟶ ∃ x10 . and (x10 ∈ x0) (x3 x9 x10 = x1).
Assume H11: ∀ x9 . ... ⟶ ∀ x10 . x10 ∈ x0 ⟶ x4 x9 x10 ∈ x0.