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.

Assume H0: explicit_OrderedField x0 x1 x2 x3 x4 x5.

Assume H1: ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ lt x0 x1 x2 x3 x4 x5 x1 x6 ⟶ x5 x1 x7 ⟶ ∃ x8 . and (prim1 x8 (1216a.. x0 (λ x9 . natOfOrderedField_p x0 x1 x2 x3 x4 x5 x9))) (x5 x7 (x4 x8 x6)).

Assume H2: ∀ x6 . prim1 x6 (b5c9f.. x0 (1216a.. x0 (λ x7 . natOfOrderedField_p x0 x1 x2 x3 x4 x5 x7))) ⟶ ∀ x7 . prim1 x7 (b5c9f.. x0 (1216a.. x0 (λ x8 . natOfOrderedField_p x0 x1 x2 x3 x4 x5 x8))) ⟶ (∀ x8 . prim1 x8 (1216a.. x0 (λ x9 . natOfOrderedField_p x0 x1 x2 x3 x4 x5 x9)) ⟶ and (and (x5 (f482f.. x6 x8) (f482f.. x7 x8)) (x5 (f482f.. x6 x8) (f482f.. x6 (x3 x8 x2)))) (x5 (f482f.. x7 (x3 x8 x2)) (f482f.. x7 x8))) ⟶ ∃ x8 . and (prim1 x8 x0) (∀ x9 . prim1 x9 (1216a.. x0 (λ x10 . natOfOrderedField_p x0 x1 x2 x3 x4 x5 x10)) ⟶ and (x5 (f482f.. x6 x9) x8) (x5 x8 (f482f.. x7 x9))).

Apply and3I with explicit_OrderedField x0 x1 x2 x3 x4 x5, ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ lt x0 x1 x2 x3 x4 x5 x1 x6 ⟶ x5 x1 x7 ⟶ ∃ x8 . and (prim1 x8 (1216a.. x0 (λ x9 . natOfOrderedField_p x0 x1 x2 x3 x4 x5 x9))) (x5 x7 (x4 x8 x6)), ∀ x6 . ... ⟶ ∀ x7 . ... ⟶ (∀ x8 . ... ⟶ and (and (x5 (f482f.. x6 x8) (f482f.. x7 x8)) (x5 (f482f.. x6 x8) (f482f.. x6 (x3 x8 x2)))) ...) ⟶ ∃ x8 . and (prim1 x8 x0) (∀ x9 . prim1 x9 (1216a.. x0 (λ x10 . natOfOrderedField_p x0 x1 x2 x3 x4 x5 x10)) ⟶ and (x5 (f482f.. x6 x9) x8) (x5 x8 (f482f.. x7 x9))) leaving 3 subgoals.

...

■

Proofgold Proof