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_Reals_E with
x0,
x1,
x2,
x3,
x4,
x5,
explicit_Reals x0 x1 x2 x6 x7 x8.
Apply explicit_OrderedField_E with
x0,
x1,
x2,
x3,
x4,
x5,
(∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ lt x0 x1 x2 x3 x4 x5 x1 x9 ⟶ x5 x1 x10 ⟶ ∃ x11 . and (x11 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) (x5 x10 (x4 x11 x9))) ⟶ (∀ x9 . x9 ∈ setexp x0 (Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ ∀ x10 . x10 ∈ setexp x0 (Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ (∀ x11 . x11 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5) ⟶ and (and (x5 (ap x9 x11) (ap x10 x11)) (x5 (ap x9 x11) (ap x9 (x3 x11 x2)))) (x5 (ap x10 (x3 x11 x2)) (ap x10 x11))) ⟶ ∃ x11 . and (x11 ∈ x0) (∀ x12 . x12 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5) ⟶ and (x5 (ap x9 x12) x11) (x5 x11 (ap x10 x12)))) ⟶ explicit_Reals x0 x1 x2 x6 x7 x8.
Apply explicit_Field_E with
x0,
x1,
x2,
x3,
x4,
... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ (∀ x9 . ... ⟶ ∀ x10 . ... ⟶ (∀ x11 . ... ⟶ and ... ...) ⟶ ∃ x11 . and (x11 ∈ x0) (∀ x12 . x12 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5) ⟶ and (x5 (ap x9 x12) x11) (x5 x11 (ap x10 x12)))) ⟶ explicit_Reals x0 x1 x2 x6 x7 x8.