Proofgold Address

Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param explicit_Reals^{explicit_Reals} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ο
Param Sep^Sep : ι → (ι → ο) → ι
Param explicit_Field_minus^{explicit_Field_minus} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι
Definition bij^bij := λ x0 x1 . λ x2 : ι → ι . and (and (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1) (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4)) (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 : ο . (∀ x5 . and (x5 ∈ x0) (x2 x5 = x3) ⟶ x4) ⟶ x4)
Definition iff^iff := λ x0 x1 : ο . and (x0 ⟶ x1) (x1 ⟶ x0)
Known explicit_Reals_transfer^{explicit_Reals_transfer} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . ∀ x6 x7 x8 . ∀ x9 x10 : ι → ι → ι . ∀ x11 : ι → ι → ο . ∀ x12 : ι → ι . explicit_Reals x0 x1 x2 x3 x4 x5 ⟶ bij x0 x6 x12 ⟶ x12 x1 = x7 ⟶ x12 x2 = x8 ⟶ (∀ x13 . x13 ∈ x0 ⟶ ∀ x14 . x14 ∈ x0 ⟶ x12 (x3 x13 x14) = x9 (x12 x13) (x12 x14)) ⟶ (∀ x13 . x13 ∈ x0 ⟶ ∀ x14 . x14 ∈ x0 ⟶ x12 (x4 x13 x14) = x10 (x12 x13) (x12 x14)) ⟶ (∀ x13 . x13 ∈ x0 ⟶ ∀ x14 . x14 ∈ x0 ⟶ iff (x5 x13 x14) (x11 (x12 x13) (x12 x14))) ⟶ explicit_Reals x6 x7 x8 x9 x10 x11
Known iff_refl^iff_refl : ∀ x0 : ο . iff x0 x0
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Theorem 9de77.. : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . ∀ x6 : ι → ι → ι . ∀ x7 . (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ x6 x8 x9 = x6 x10 x11 ⟶ and (x8 = x10) (x9 = x11)) ⟶ explicit_Reals x0 x1 x2 x3 x4 x5 ⟶ (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x3 x8 x9 = x3 x9 x8) ⟶ x1 ∈ x0 ⟶ (∀ x8 . x8 ∈ x0 ⟶ x3 x1 x8 = x8) ⟶ (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x4 x8 x9 ∈ x0) ⟶ (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ prim0 (λ x11 . and (x11 ∈ x0) (∀ x12 : ο . (∀ x13 . and (x13 ∈ x0) (x6 x8 x9 = x6 x11 x13) ⟶ x12) ⟶ x12)) = x8) ⟶ (∀ x8 . x8 ∈ x0 ⟶ x6 x8 x1 ∈ {x9 ∈ x7|x6 (prim0 (λ x11 . and (x11 ∈ x0) (∀ x12 : ο . (∀ x13 . and (x13 ∈ x0) (x9 = x6 x11 x13) ⟶ x12) ⟶ x12))) x1 = x9}) ⟶ (∀ x8 . x8 ∈ x7 ⟶ prim0 (λ x9 . and (x9 ∈ x0) (∀ x10 : ο . (∀ x11 . and (x11 ∈ x0) (x8 = x6 x9 x11) ⟶ x10) ⟶ x10)) ∈ x0) ⟶ (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ x6 (x3 (prim0 (λ x13 . and (x13 ∈ x0) (∀ x14 : ο . (∀ x15 . and (x15 ∈ x0) (x6 x8 x9 = x6 x13 x15) ⟶ x14) ⟶ x14))) (prim0 (λ x13 . and (x13 ∈ x0) (∀ x14 : ο . (∀ x15 . and (x15 ∈ x0) (x6 x10 x11 = x6 x13 x15) ⟶ x14) ⟶ x14)))) (x3 (prim0 (λ x13 . and (x13 ∈ x0) (x6 x8 x9 = x6 (prim0 (λ x15 . and (x15 ∈ x0) (∀ x16 : ο . (∀ x17 . and (x17 ∈ x0) (x6 x8 x9 = x6 x15 x17) ⟶ x16) ⟶ x16))) x13))) (prim0 (λ x13 . and (x13 ∈ x0) (x6 x10 x11 = x6 (prim0 (λ x15 . and (x15 ∈ x0) (∀ x16 : ο . (∀ x17 . and (x17 ∈ x0) (x6 x10 x11 = x6 x15 x17) ⟶ x16) ⟶ x16))) x13)))) = x6 (x3 x8 x10) (x3 x9 x11)) ⟶ (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ x6 (x3 (x4 (prim0 (λ x13 . and (x13 ∈ x0) (∀ x14 : ο . (∀ x15 . and (x15 ∈ x0) (x6 x8 x9 = x6 x13 x15) ⟶ x14) ⟶ x14))) (prim0 (λ x13 . and (x13 ∈ x0) (∀ x14 : ο . (∀ x15 . and (x15 ∈ x0) (x6 x10 x11 = x6 x13 x15) ⟶ x14) ⟶ x14)))) (explicit_Field_minus x0 x1 x2 x3 x4 (x4 (prim0 (λ x13 . and (x13 ∈ x0) (x6 x8 x9 = x6 (prim0 (λ x15 . and (x15 ∈ x0) (∀ x16 : ο . (∀ x17 . and (x17 ∈ x0) (x6 x8 x9 = x6 x15 x17) ⟶ x16) ⟶ x16))) x13))) (prim0 (λ x13 . and (x13 ∈ x0) (x6 x10 x11 = x6 (prim0 (λ x15 . and (x15 ∈ x0) (∀ x16 : ο . (∀ x17 . and (x17 ∈ x0) (x6 x10 x11 = x6 x15 x17) ⟶ x16) ⟶ x16))) x13)))))) (x3 (x4 (prim0 (λ x13 . and (x13 ∈ x0) (∀ x14 : ο . (∀ x15 . and (x15 ∈ x0) (x6 x8 x9 = x6 x13 x15) ⟶ x14) ⟶ x14))) (prim0 (λ x13 . and (x13 ∈ x0) (x6 x10 x11 = x6 (prim0 (λ x15 . and (x15 ∈ x0) (∀ x16 : ο . (∀ x17 . and (x17 ∈ x0) (x6 x10 x11 = x6 x15 x17) ⟶ x16) ⟶ x16))) x13)))) (x4 (prim0 (λ x13 . and (x13 ∈ x0) (x6 x8 x9 = x6 (prim0 (λ x15 . and (x15 ∈ x0) (∀ x16 : ο . (∀ x17 . and (x17 ∈ x0) (x6 x8 x9 = x6 x15 x17) ⟶ x16) ⟶ x16))) x13))) (prim0 (λ x13 . and (x13 ∈ x0) (∀ x14 : ο . (∀ x15 . and (x15 ∈ x0) (x6 x10 x11 = x6 x13 x15) ⟶ x14) ⟶ x14))))) = x6 (x3 (x4 x8 x10) (explicit_Field_minus x0 x1 x2 x3 x4 (x4 x9 x11))) (x3 (x4 x8 x11) (x4 x9 x10))) ⟶ explicit_Field_minus x0 x1 x2 x3 x4 x1 = x1 ⟶ (∀ x8 . x8 ∈ x0 ⟶ x4 x1 x8 = x1) ⟶ (∀ x8 . x8 ∈ x0 ⟶ x4 x8 x1 = x1) ⟶ explicit_Reals {x8 ∈ x7|x6 (prim0 (λ x10 . and (x10 ∈ x0) (∀ x11 : ο . (∀ x12 . and (x12 ∈ x0) (x8 = x6 x10 x12) ⟶ x11) ⟶ x11))) x1 = x8} (x6 x1 x1) (x6 x2 x1) (λ x8 x9 . x6 (x3 (prim0 (λ x10 . and (x10 ∈ x0) (∀ x11 : ο . (∀ x12 . and (x12 ∈ x0) (x8 = x6 x10 x12) ⟶ x11) ⟶ x11))) (prim0 (λ x10 . and (x10 ∈ x0) (∀ x11 : ο . (∀ x12 . and (x12 ∈ x0) (x9 = x6 x10 x12) ⟶ x11) ⟶ x11)))) (x3 (prim0 (λ x10 . and (x10 ∈ x0) (x8 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∀ x13 : ο . (∀ x14 . and (x14 ∈ x0) (x8 = x6 x12 x14) ⟶ x13) ⟶ x13))) x10))) (prim0 (λ x10 . and (x10 ∈ x0) (x9 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∀ x13 : ο . (∀ x14 . and (x14 ∈ x0) (x9 = x6 x12 x14) ⟶ x13) ⟶ x13))) x10))))) (λ x8 x9 . x6 (x3 (x4 (prim0 (λ x10 . and (x10 ∈ x0) (∀ x11 : ο . (∀ x12 . and (x12 ∈ x0) (x8 = x6 x10 x12) ⟶ x11) ⟶ x11))) (prim0 (λ x10 . and (x10 ∈ x0) (∀ x11 : ο . (∀ x12 . and (x12 ∈ x0) (x9 = x6 x10 x12) ⟶ x11) ⟶ x11)))) (explicit_Field_minus x0 x1 x2 x3 x4 (x4 (prim0 (λ x10 . and (x10 ∈ x0) (x8 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∀ x13 : ο . (∀ x14 . and (x14 ∈ x0) (x8 = x6 x12 x14) ⟶ x13) ⟶ x13))) x10))) (prim0 (λ x10 . and (x10 ∈ x0) (x9 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∀ x13 : ο . (∀ x14 . and (x14 ∈ x0) (x9 = x6 x12 x14) ⟶ x13) ⟶ x13))) x10)))))) (x3 (x4 (prim0 (λ x10 . and (x10 ∈ x0) (∀ x11 : ο . (∀ x12 . and (x12 ∈ x0) (x8 = x6 x10 x12) ⟶ x11) ⟶ x11))) (prim0 (λ x10 . and (x10 ∈ x0) (x9 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∀ x13 : ο . (∀ x14 . and (x14 ∈ x0) (x9 = x6 x12 x14) ⟶ x13) ⟶ x13))) x10)))) (x4 (prim0 (λ x10 . and (x10 ∈ x0) (x8 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∀ x13 : ο . (∀ x14 . and (x14 ∈ x0) (x8 = x6 x12 x14) ⟶ x13) ⟶ x13))) x10))) (prim0 (λ x10 . and (x10 ∈ x0) (∀ x11 : ο . (∀ x12 . and (x12 ∈ x0) (x9 = x6 x10 x12) ⟶ x11) ⟶ x11)))))) (λ x8 x9 . x5 (prim0 (λ x10 . and (x10 ∈ x0) (∀ x11 : ο . (∀ x12 . and (x12 ∈ x0) (x8 = x6 x10 x12) ⟶ x11) ⟶ x11))) (prim0 (λ x10 . and (x10 ∈ x0) (∀ x11 : ο . (∀ x12 . and (x12 ∈ x0) (x9 = x6 x10 x12) ⟶ x11) ⟶ x11)))) (proof)