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 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ lt x0 x1 x2 x3 x4 x5 x1 x6 ⟶ x5 x1 x7 ⟶ ∃ x8 . and (x8 ∈ {x9 ∈ x0|natOfOrderedField_p x0 x1 x2 x3 x4 x5 x9}) (x5 x7 (x4 x8 x6)).

Assume H2: ∀ x6 . x6 ∈ setexp x0 {x7 ∈ x0|natOfOrderedField_p x0 x1 x2 x3 x4 x5 x7} ⟶ ∀ x7 . x7 ∈ setexp x0 {x8 ∈ x0|natOfOrderedField_p x0 x1 x2 x3 x4 x5 x8} ⟶ (∀ x8 . x8 ∈ {x9 ∈ x0|natOfOrderedField_p x0 x1 x2 x3 x4 x5 x9} ⟶ 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 ∈ {x10 ∈ x0|natOfOrderedField_p x0 x1 x2 x3 x4 x5 x10} ⟶ and (x5 (ap x6 x9) x8) (x5 x8 (ap x7 x9))).

Apply and3I with explicit_OrderedField 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 ∈ {x9 ∈ x0|natOfOrderedField_p x0 x1 x2 x3 x4 x5 x9}) (x5 x7 (x4 x8 x6)), ∀ x6 . ... ⟶ ∀ x7 . ... ⟶ (∀ x8 . x8 ∈ {x9 ∈ x0|natOfOrderedField_p x0 x1 x2 x3 ... ... ...} ⟶ 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 ∈ {x10 ∈ x0|natOfOrderedField_p x0 x1 x2 x3 x4 x5 x10} ⟶ and (x5 (ap x6 x9) x8) (x5 x8 (ap x7 x9))) leaving 3 subgoals.

...

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Proofgold Proof