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,
explicit_Field x0 x1 x2 x3 x4.
Assume H6: ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x5 x6 x7 ⟶ x5 x7 x8 ⟶ x5 x6 x8.
Assume H7:
∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ iff (and (x5 x6 x7) (x5 x7 x6)) (x6 = x7).
Assume H8:
∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ or (x5 x6 x7) (x5 x7 x6).
Assume H9: ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x5 x6 x7 ⟶ x5 (x3 x6 x8) (x3 x7 x8).
Assume H10: ∀ x6 . ... ⟶ ∀ x7 . x7 ∈ x0 ⟶ x5 x1 x6 ⟶ x5 x1 x7 ⟶ x5 x1 (x4 x6 x7).
Apply set_ext with
{x6 ∈ x0|c3146.. x0 x1 x2 x3 x4 x5 x6},
x0 leaving 2 subgoals.
The subproof is completed by applying Sep_Subq with
x0,
λ x6 . c3146.. x0 x1 x2 x3 x4 x5 x6.
Let x6 of type ι be given.
Assume H5: x6 ∈ x0.
Apply SepI with
x0,
c3146.. x0 x1 x2 x3 x4 x5,
x6 leaving 2 subgoals.
The subproof is completed by applying H5.
Let x7 of type ο be given.
Apply H6 with
x1.
Let x8 of type ο be given.
Apply H7 with
x1.
Apply and3I with
natOfOrderedField_p x0 x1 x2 x3 x4 x5 x1,
natOfOrderedField_p x0 x1 x2 x3 x4 x5 x1,
x4 x1 x6 = x1 leaving 3 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying L3.
Apply explicit_Field_zero_multL with
x0,
x1,
x2,
x3,
x4,
x6 leaving 2 subgoals.
Apply L4.
The subproof is completed by applying H0.
The subproof is completed by applying H5.