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.
Apply explicit_OrderedField_E with
x0,
x1,
x2,
x3,
x4,
x5,
∀ x6 . x6 ⊆ x0 ⟶ explicit_Field x6 x1 x2 x3 x4 ⟶ {x7 ∈ x0|explicit_OrderedField_rationalp x0 x1 x2 x3 x4 x5 x7} ⊆ x6.
Apply explicit_Field_E with
x0,
x1,
x2,
x3,
x4,
(∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x5 x6 x7 ⟶ x5 x7 x8 ⟶ x5 x6 x8) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ iff (and (x5 x6 x7) (x5 x7 x6)) (x6 = x7)) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ or (x5 x6 x7) (x5 x7 x6)) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x5 x6 x7 ⟶ x5 (x3 x6 x8) (x3 x7 x8)) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x5 x1 x6 ⟶ x5 x1 x7 ⟶ x5 x1 (x4 x6 x7)) ⟶ ∀ x6 . x6 ⊆ x0 ⟶ explicit_Field x6 x1 x2 x3 x4 ⟶ {x7 ∈ x0|explicit_OrderedField_rationalp x0 x1 x2 x3 x4 x5 x7} ⊆ x6.
Assume H2: ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x6 x7 ∈ x0.
Assume H3: ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x3 x6 (x3 x7 x8) = x3 (x3 x6 x7) x8.
Assume H4: ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x6 x7 = x3 x7 x6.
Assume H5: x1 ∈ x0.
Assume H6: ∀ x6 . x6 ∈ x0 ⟶ x3 x1 x6 = x6.
Assume H7:
∀ x6 . x6 ∈ x0 ⟶ ∃ x7 . and (x7 ∈ x0) (x3 x6 x7 = x1).
Assume H8: ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x6 x7 ∈ x0.
Assume H9: ∀ x6 . ... ⟶ ∀ x7 . ... ⟶ ∀ x8 . ... ∈ ... ⟶ x4 x6 (x4 x7 x8) = x4 (x4 x6 x7) x8.