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.

Let x6 of type ο be given.

Assume H0: explicit_Reals x0 x1 x2 x3 x4 x5 ⟶ explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ lt x0 x1 x2 x3 x4 x5 x1 x7 ⟶ x5 x1 x8 ⟶ ∃ x9 . and (x9 ∈ {x10 ∈ x0|natOfOrderedField_p x0 x1 x2 x3 x4 x5 x10}) (x5 x8 (x4 x9 x7))) ⟶ (∀ x7 . x7 ∈ setexp x0 {x8 ∈ x0|natOfOrderedField_p x0 x1 x2 x3 x4 x5 x8} ⟶ ∀ x8 . x8 ∈ setexp x0 {x9 ∈ x0|natOfOrderedField_p x0 x1 x2 x3 x4 x5 x9} ⟶ (∀ x9 . x9 ∈ {x10 ∈ x0|natOfOrderedField_p x0 x1 x2 x3 x4 x5 x10} ⟶ and (and (x5 (ap x7 x9) (ap x8 x9)) (x5 (ap x7 x9) (ap x7 (x3 x9 x2)))) (x5 (ap x8 (x3 x9 x2)) (ap x8 x9))) ⟶ ∃ x9 . and (x9 ∈ x0) (∀ x10 . x10 ∈ {x11 ∈ x0|natOfOrderedField_p x0 x1 x2 x3 x4 x5 x11} ⟶ and (x5 (ap x7 x10) x9) (x5 x9 (ap x8 x10)))) ⟶ x6.

Assume H1: explicit_Reals x0 x1 x2 x3 x4 x5.

Apply and3E with explicit_OrderedField x0 x1 x2 x3 x4 x5, ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ lt x0 x1 x2 x3 x4 x5 x1 x7 ⟶ x5 x1 x8 ⟶ ∃ x9 . and (x9 ∈ {x10 ∈ x0|natOfOrderedField_p x0 x1 x2 x3 x4 x5 x10}) (x5 x8 (x4 x9 x7)), ∀ x7 . ... ⟶ ∀ x8 . ... ⟶ (∀ x9 . ... ⟶ and (and (x5 (ap x7 x9) (ap x8 x9)) (x5 (ap x7 x9) (ap x7 (x3 x9 x2)))) (x5 (ap ... ...) ...)) ⟶ ∃ x9 . and (x9 ∈ x0) (∀ x10 . x10 ∈ {x11 ∈ x0|natOfOrderedField_p x0 x1 x2 x3 x4 x5 x11} ⟶ and (x5 (ap x7 x10) x9) (x5 x9 (ap x8 x10))), ... leaving 2 subgoals.

...

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Proofgold Proof