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 ⟶ x5 x6 x6.
Assume H2: ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x5 x6 x7 ⟶ x5 x7 x8 ⟶ x5 x6 x8.
Assume H3:
∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ iff (and (x5 x6 x7) (x5 x7 x6)) (x6 = x7).
Assume H4:
∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ or (x5 x6 x7) (x5 x7 x6).
Assume H5: ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x5 x6 x7 ⟶ x5 (x3 x6 x8) (x3 x7 x8).
Assume H6: ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x5 x1 x6 ⟶ x5 x1 x7 ⟶ x5 x1 (x4 x6 x7).
Let x6 of type ι be given.
Assume H7: x6 ∈ x0.
Apply H3 with
x6,
x6,
x5 x6 x6 leaving 3 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H7.
Assume H8:
and (x5 x6 x6) (x5 x6 x6) ⟶ x6 = x6.
Assume H9:
x6 = x6 ⟶ and (x5 x6 x6) (x5 x6 x6).
Apply H9 with
x5 x6 x6 leaving 2 subgoals.
Let x7 of type ι → ι → ο be given.
Assume H10: x7 x6 x6.
The subproof is completed by applying H10.
Assume H10: x5 x6 x6.
Assume H11: x5 x6 x6.
The subproof is completed by applying H10.