Proofgold Address

Definition and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param 62ee1.. : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ο
Param 1216a.. : ι → (ι → ο) → ι
Param explicit_Field_minus : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι
Definition bij := λ x0 x1 . λ x2 : ι → ι . and (and (∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) x1) (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4)) (∀ x3 . prim1 x3 x1 ⟶ ∃ x4 . and (prim1 x4 x0) (x2 x4 = x3))
Definition iff := λ x0 x1 : ο . and (x0 ⟶ x1) (x1 ⟶ x0)
Known 5ece9.. : ∀ 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 . 62ee1.. x0 x1 x2 x3 x4 x5 ⟶ bij x0 x6 x12 ⟶ x12 x1 = x7 ⟶ x12 x2 = x8 ⟶ (∀ x13 . prim1 x13 x0 ⟶ ∀ x14 . prim1 x14 x0 ⟶ x12 (x3 x13 x14) = x9 (x12 x13) (x12 x14)) ⟶ (∀ x13 . prim1 x13 x0 ⟶ ∀ x14 . prim1 x14 x0 ⟶ x12 (x4 x13 x14) = x10 (x12 x13) (x12 x14)) ⟶ (∀ x13 . prim1 x13 x0 ⟶ ∀ x14 . prim1 x14 x0 ⟶ iff (x5 x13 x14) (x11 (x12 x13) (x12 x14))) ⟶ 62ee1.. x6 x7 x8 x9 x10 x11
Known iff_refl : ∀ x0 : ο . iff x0 x0
Known and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known 492ff.. : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . prim1 x2 (1216a.. x0 x1) ⟶ ∀ x3 : ο . (prim1 x2 x0 ⟶ x1 x2 ⟶ x3) ⟶ x3
Known andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Theorem b6a82.. : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . ∀ x6 : ι → ι → ι . ∀ x7 . (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ ∀ x10 . prim1 x10 x0 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x6 x8 x9 = x6 x10 x11 ⟶ and (x8 = x10) (x9 = x11)) ⟶ 62ee1.. x0 x1 x2 x3 x4 x5 ⟶ (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x3 x8 x9 = x3 x9 x8) ⟶ prim1 x1 x0 ⟶ (∀ x8 . prim1 x8 x0 ⟶ x3 x1 x8 = x8) ⟶ (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ prim1 (x4 x8 x9) x0) ⟶ (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ prim0 (λ x11 . and (prim1 x11 x0) (∃ x12 . and (prim1 x12 x0) (x6 x8 x9 = x6 x11 x12))) = x8) ⟶ (∀ x8 . prim1 x8 x0 ⟶ prim1 (x6 x8 x1) (1216a.. x7 (λ x9 . x6 (prim0 (λ x11 . and (prim1 x11 x0) (∃ x12 . and (prim1 x12 x0) (x9 = x6 x11 x12)))) x1 = x9))) ⟶ (∀ x8 . prim1 x8 x7 ⟶ prim1 (prim0 (λ x9 . and (prim1 x9 x0) (∃ x10 . and (prim1 x10 x0) (x8 = x6 x9 x10)))) x0) ⟶ (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ ∀ x10 . prim1 x10 x0 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x6 (x3 (prim0 (λ x13 . and (prim1 x13 x0) (∃ x14 . and (prim1 x14 x0) (x6 x8 x9 = x6 x13 x14)))) (prim0 (λ x13 . and (prim1 x13 x0) (∃ x14 . and (prim1 x14 x0) (x6 x10 x11 = x6 x13 x14))))) (x3 (prim0 (λ x13 . and (prim1 x13 x0) (x6 x8 x9 = x6 (prim0 (λ x15 . and (prim1 x15 x0) (∃ x16 . and (prim1 x16 x0) (x6 x8 x9 = x6 x15 x16)))) x13))) (prim0 (λ x13 . and (prim1 x13 x0) (x6 x10 x11 = x6 (prim0 (λ x15 . and (prim1 x15 x0) (∃ x16 . and (prim1 x16 x0) (x6 x10 x11 = x6 x15 x16)))) x13)))) = x6 (x3 x8 x10) (x3 x9 x11)) ⟶ (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ ∀ x10 . prim1 x10 x0 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x6 (x3 (x4 (prim0 (λ x13 . and (prim1 x13 x0) (∃ x14 . and (prim1 x14 x0) (x6 x8 x9 = x6 x13 x14)))) (prim0 (λ x13 . and (prim1 x13 x0) (∃ x14 . and (prim1 x14 x0) (x6 x10 x11 = x6 x13 x14))))) (explicit_Field_minus x0 x1 x2 x3 x4 (x4 (prim0 (λ x13 . and (prim1 x13 x0) (x6 x8 x9 = x6 (prim0 (λ x15 . and (prim1 x15 x0) (∃ x16 . and (prim1 x16 x0) (x6 x8 x9 = x6 x15 x16)))) x13))) (prim0 (λ x13 . and (prim1 x13 x0) (x6 x10 x11 = x6 (prim0 (λ x15 . and (prim1 x15 x0) (∃ x16 . and (prim1 x16 x0) (x6 x10 x11 = x6 x15 x16)))) x13)))))) (x3 (x4 (prim0 (λ x13 . and (prim1 x13 x0) (∃ x14 . and (prim1 x14 x0) (x6 x8 x9 = x6 x13 x14)))) (prim0 (λ x13 . and (prim1 x13 x0) (x6 x10 x11 = x6 (prim0 (λ x15 . and (prim1 x15 x0) (∃ x16 . and (prim1 x16 x0) (x6 x10 x11 = x6 x15 x16)))) x13)))) (x4 (prim0 (λ x13 . and (prim1 x13 x0) (x6 x8 x9 = x6 (prim0 (λ x15 . and (prim1 x15 x0) (∃ x16 . and (prim1 x16 x0) (x6 x8 x9 = x6 x15 x16)))) x13))) (prim0 (λ x13 . and (prim1 x13 x0) (∃ x14 . and (prim1 x14 x0) (x6 x10 x11 = x6 x13 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 . prim1 x8 x0 ⟶ x4 x1 x8 = x1) ⟶ (∀ x8 . prim1 x8 x0 ⟶ x4 x8 x1 = x1) ⟶ 62ee1.. (1216a.. x7 (λ x8 . x6 (prim0 (λ x10 . and (prim1 x10 x0) (∃ x11 . and (prim1 x11 x0) (x8 = x6 x10 x11)))) x1 = x8)) (x6 x1 x1) (x6 x2 x1) (λ x8 x9 . x6 (x3 (prim0 (λ x10 . and (prim1 x10 x0) (∃ x11 . and (prim1 x11 x0) (x8 = x6 x10 x11)))) (prim0 (λ x10 . and (prim1 x10 x0) (∃ x11 . and (prim1 x11 x0) (x9 = x6 x10 x11))))) (x3 (prim0 (λ x10 . and (prim1 x10 x0) (x8 = x6 (prim0 (λ x12 . and (prim1 x12 x0) (∃ x13 . and (prim1 x13 x0) (x8 = x6 x12 x13)))) x10))) (prim0 (λ x10 . and (prim1 x10 x0) (x9 = x6 (prim0 (λ x12 . and (prim1 x12 x0) (∃ x13 . and (prim1 x13 x0) (x9 = x6 x12 x13)))) x10))))) (λ x8 x9 . x6 (x3 (x4 (prim0 (λ x10 . and (prim1 x10 x0) (∃ x11 . and (prim1 x11 x0) (x8 = x6 x10 x11)))) (prim0 (λ x10 . and (prim1 x10 x0) (∃ x11 . and (prim1 x11 x0) (x9 = x6 x10 x11))))) (explicit_Field_minus x0 x1 x2 x3 x4 (x4 (prim0 (λ x10 . and (prim1 x10 x0) (x8 = x6 (prim0 (λ x12 . and (prim1 x12 x0) (∃ x13 . and (prim1 x13 x0) (x8 = x6 x12 x13)))) x10))) (prim0 (λ x10 . and (prim1 x10 x0) (x9 = x6 (prim0 (λ x12 . and (prim1 x12 x0) (∃ x13 . and (prim1 x13 x0) (x9 = x6 x12 x13)))) x10)))))) (x3 (x4 (prim0 (λ x10 . and (prim1 x10 x0) (∃ x11 . and (prim1 x11 x0) (x8 = x6 x10 x11)))) (prim0 (λ x10 . and (prim1 x10 x0) (x9 = x6 (prim0 (λ x12 . and (prim1 x12 x0) (∃ x13 . and (prim1 x13 x0) (x9 = x6 x12 x13)))) x10)))) (x4 (prim0 (λ x10 . and (prim1 x10 x0) (x8 = x6 (prim0 (λ x12 . and (prim1 x12 x0) (∃ x13 . and (prim1 x13 x0) (x8 = x6 x12 x13)))) x10))) (prim0 (λ x10 . and (prim1 x10 x0) (∃ x11 . and (prim1 x11 x0) (x9 = x6 x10 x11))))))) (λ x8 x9 . x5 (prim0 (λ x10 . and (prim1 x10 x0) (∃ x11 . and (prim1 x11 x0) (x8 = x6 x10 x11)))) (prim0 (λ x10 . and (prim1 x10 x0) (∃ x11 . and (prim1 x11 x0) (x9 = x6 x10 x11)))))