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_Reals_E with
x0,
x1,
x2,
x3,
x4,
x5,
∀ x6 . x6 ∈ omega ⟶ nat_primrec x2 (λ x7 x8 . x3 x2 x8) x6 = x1 ⟶ ∀ x7 : ο . x7.
Apply explicit_OrderedField_E with
x0,
x1,
x2,
x3,
x4,
x5,
(∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ lt x0 x1 x2 x3 x4 x5 x1 x6 ⟶ x5 x1 x7 ⟶ ∃ x8 . and (x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) (x5 x7 (x4 x8 x6))) ⟶ (∀ x6 . x6 ∈ setexp x0 (Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ ∀ x7 . x7 ∈ setexp x0 (Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ (∀ x8 . x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5) ⟶ and (and (x5 (ap x6 x8) (ap x7 x8)) (x5 (ap x6 x8) (ap x6 (x3 x8 x2)))) (x5 (ap x7 (x3 x8 x2)) (ap x7 x8))) ⟶ ∃ x8 . and (x8 ∈ x0) (∀ x9 . x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5) ⟶ and (x5 (ap x6 x9) x8) (x5 x8 (ap x7 x9)))) ⟶ ∀ x6 . x6 ∈ omega ⟶ nat_primrec x2 (λ x7 x8 . x3 x2 x8) x6 = x1 ⟶ ∀ x7 : ο . x7.
Apply explicit_Field_E with
x0,
x1,
x2,
x3,
x4,
... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ (∀ x6 . ... ⟶ ∀ x7 . ... ⟶ (∀ x8 . ... ⟶ and (and (x5 (ap x6 x8) (ap x7 x8)) (x5 (ap x6 x8) (ap x6 (x3 x8 x2)))) (x5 (ap x7 (x3 x8 x2)) ...)) ⟶ ∃ x8 . and (x8 ∈ x0) (∀ x9 . x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5) ⟶ and (x5 (ap x6 x9) x8) (x5 x8 (ap x7 x9)))) ⟶ ∀ x6 . x6 ∈ omega ⟶ nat_primrec x2 (λ x7 x8 . x3 x2 x8) x6 = x1 ⟶ ∀ x7 : ο . x7.