Proofgold Address

Definition False := ∀ x0 : ο . x0
Definition True := ∀ x0 : ο . x0 ⟶ x0
Definition not := λ x0 : ο . x0 ⟶ False
Theorem FalseE : False ⟶ ∀ x0 : ο . x0 (proof)
Theorem TrueI : True (proof)
Theorem notI : ∀ x0 : ο . (x0 ⟶ False) ⟶ not x0 (proof)
Theorem notE : ∀ x0 : ο . not x0 ⟶ x0 ⟶ False (proof)
Definition and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Theorem andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1 (proof)
Theorem andEL : ∀ x0 x1 : ο . and x0 x1 ⟶ x0 (proof)
Theorem andER : ∀ x0 x1 : ο . and x0 x1 ⟶ x1 (proof)
Definition or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Theorem orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1 (proof)
Theorem orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1 (proof)
Theorem orE : ∀ x0 x1 x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ or x0 x1 ⟶ x2 (proof)
Definition iff := λ x0 x1 : ο . and (x0 ⟶ x1) (x1 ⟶ x0)
Theorem iffEL : ∀ x0 x1 : ο . iff x0 x1 ⟶ x0 ⟶ x1 (proof)
Theorem iffER : ∀ x0 x1 : ο . iff x0 x1 ⟶ x1 ⟶ x0 (proof)
Theorem iffI : ∀ x0 x1 : ο . (x0 ⟶ x1) ⟶ (x1 ⟶ x0) ⟶ iff x0 x1 (proof)
Theorem iff_refl : ∀ x0 : ο . iff x0 x0 (proof)
Known prop_ext : ∀ x0 x1 : ο . iff x0 x1 ⟶ x0 = x1
Definition nIn := λ x0 x1 . not (prim1 x0 x1)
Definition Subq := λ x0 x1 . ∀ x2 . prim1 x2 x0 ⟶ prim1 x2 x1
Definition 91630.. := λ x0 . prim2 x0 x0
Definition 7ee77.. := λ x0 x1 . prim2 (prim2 x0 x1) (91630.. x0)
Definition c2e41.. := λ x0 x1 . ∀ x2 : ο . (∀ x3 . and (and (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 : ο . (∀ x6 . and (prim1 x6 x1) (prim1 (7ee77.. x4 x6) x3) ⟶ x5) ⟶ x5) (∀ x4 . prim1 x4 x1 ⟶ ∀ x5 : ο . (∀ x6 . and (prim1 x6 x0) (prim1 (7ee77.. x6 x4) x3) ⟶ x5) ⟶ x5)) (∀ x4 x5 x6 x7 . prim1 (7ee77.. x4 x5) x3 ⟶ prim1 (7ee77.. x6 x7) x3 ⟶ iff (x4 = x6) (x5 = x7)) ⟶ x2) ⟶ x2
Known Eps_i_ax : ∀ x0 : ι → ο . ∀ x1 . x0 x1 ⟶ x0 (prim0 x0)
Known 0ddd1.. : ∀ x0 x1 . (∀ x2 . iff (prim1 x2 x0) (prim1 x2 x1)) ⟶ x0 = x1
Known 53c21.. : ∀ x0 x1 x2 . iff (prim1 x0 (prim2 x1 x2)) (or (x0 = x1) (x0 = x2))
Known UnionEq : ∀ x0 x1 . iff (prim1 x1 (prim3 x0)) (∀ x2 : ο . (∀ x3 . and (prim1 x1 x3) (prim1 x3 x0) ⟶ x2) ⟶ x2)
Known e8b3c.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 x3 x4 . x1 x2 x3 ⟶ x1 x2 x4 ⟶ x3 = x4) ⟶ ∀ x2 : ο . (∀ x3 . (∀ x4 . iff (prim1 x4 x3) (∀ x5 : ο . (∀ x6 . and (prim1 x6 x0) (x1 x6 x4) ⟶ x5) ⟶ x5)) ⟶ x2) ⟶ x2
Theorem Eps_i_ex : ∀ x0 : ι → ο . (∀ x1 : ο . (∀ x2 . x0 x2 ⟶ x1) ⟶ x1) ⟶ x0 (prim0 x0) (proof)
Theorem pred_ext : ∀ x0 x1 : ι → ο . (∀ x2 . iff (x0 x2) (x1 x2)) ⟶ x0 = x1 (proof)
Theorem prop_ext_2 : ∀ x0 x1 : ο . (x0 ⟶ x1) ⟶ (x1 ⟶ x0) ⟶ x0 = x1 (proof)
Theorem pred_ext_2 : ∀ x0 x1 : ι → ο . (∀ x2 . x0 x2 ⟶ x1 x2) ⟶ (∀ x2 . x1 x2 ⟶ x0 x2) ⟶ x0 = x1 (proof)
Theorem 2532b.. : ∀ x0 x1 x2 . prim1 x0 (prim2 x1 x2) ⟶ or (x0 = x1) (x0 = x2) (proof)
Theorem 67787.. : ∀ x0 x1 . prim1 x0 (prim2 x0 x1) (proof)
Theorem 5a932.. : ∀ x0 x1 . prim1 x1 (prim2 x0 x1) (proof)
Theorem e7a4c.. : ∀ x0 . prim1 x0 (91630.. x0) (proof)
Theorem fead7.. : ∀ x0 x1 . prim1 x1 (91630.. x0) ⟶ x1 = x0 (proof)
Theorem 0117f.. : ∀ x0 : ο . (∀ x1 . (∀ x2 . nIn x2 x1) ⟶ x0) ⟶ x0 (proof)
Definition 4a7ef.. := prim0 (λ x0 . ∀ x1 . nIn x1 x0)
Theorem dcd83.. : ∀ x0 . nIn x0 4a7ef.. (proof)
Theorem xm : ∀ x0 : ο . or x0 (not x0) (proof)
Theorem UnionI : ∀ x0 x1 x2 . prim1 x1 x2 ⟶ prim1 x2 x0 ⟶ prim1 x1 (prim3 x0) (proof)
Theorem UnionE : ∀ x0 x1 . prim1 x1 (prim3 x0) ⟶ ∀ x2 : ο . (∀ x3 . and (prim1 x1 x3) (prim1 x3 x0) ⟶ x2) ⟶ x2 (proof)
Theorem UnionE_impred : ∀ x0 x1 . prim1 x1 (prim3 x0) ⟶ ∀ x2 : ο . (∀ x3 . prim1 x1 x3 ⟶ prim1 x3 x0 ⟶ x2) ⟶ x2 (proof)
Definition c2f57.. := λ x0 : ι → ο . ∀ x1 : ο . (∀ x2 . (∀ x3 . iff (prim1 x3 x2) (x0 x3)) ⟶ x1) ⟶ x1
Definition 707e2.. := λ x0 : ι → ο . prim0 (λ x1 . ∀ x2 . iff (prim1 x2 x1) (x0 x2))
Theorem 8098c.. : ∀ x0 : ι → ο . c2f57.. x0 ⟶ ∀ x1 . iff (prim1 x1 (707e2.. x0)) (x0 x1) (proof)
Theorem bbc77.. : ∀ x0 : ι → ο . c2f57.. x0 ⟶ ∀ x1 . x0 x1 ⟶ prim1 x1 (707e2.. x0) (proof)
Theorem 477e8.. : ∀ x0 : ι → ο . c2f57.. x0 ⟶ ∀ x1 . prim1 x1 (707e2.. x0) ⟶ x0 x1 (proof)
Theorem b2728.. : ∀ x0 . c2f57.. (λ x1 . prim1 x1 x0) (proof)
Theorem 71e64.. : ∀ x0 . 707e2.. (λ x2 . prim1 x2 x0) = x0 (proof)
Theorem f336f.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ο . (∀ x3 . (∀ x4 . iff (prim1 x4 x3) (∀ x5 : ο . (∀ x6 . and (prim1 x6 x0) (x4 = x1 x6) ⟶ x5) ⟶ x5)) ⟶ x2) ⟶ x2 (proof)
Definition 94f9e.. := λ x0 . λ x1 : ι → ι . prim0 (λ x2 . ∀ x3 . iff (prim1 x3 x2) (∀ x4 : ο . (∀ x5 . and (prim1 x5 x0) (x3 = x1 x5) ⟶ x4) ⟶ x4))
Theorem b81d7.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . iff (prim1 x2 (94f9e.. x0 x1)) (∀ x3 : ο . (∀ x4 . and (prim1 x4 x0) (x2 = x1 x4) ⟶ x3) ⟶ x3) (proof)
Theorem 696c0.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 x0 ⟶ prim1 (x1 x2) (94f9e.. x0 x1) (proof)
Theorem 6acac.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 (94f9e.. x0 x1) ⟶ ∀ x3 : ο . (∀ x4 . and (prim1 x4 x0) (x2 = x1 x4) ⟶ x3) ⟶ x3 (proof)
Theorem 8c6f6.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 (94f9e.. x0 x1) ⟶ ∀ x3 : ο . (∀ x4 . prim1 x4 x0 ⟶ x2 = x1 x4 ⟶ x3) ⟶ x3 (proof)
Theorem 15fbb.. : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 : ο . (∀ x3 . (∀ x4 . iff (prim1 x4 x3) (and (prim1 x4 x0) (x1 x4))) ⟶ x2) ⟶ x2 (proof)
Definition 1216a.. := λ x0 . λ x1 : ι → ο . prim0 (λ x2 . ∀ x3 . iff (prim1 x3 x2) (and (prim1 x3 x0) (x1 x3)))
Theorem 92823.. : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . iff (prim1 x2 (1216a.. x0 x1)) (and (prim1 x2 x0) (x1 x2)) (proof)
Theorem b2421.. : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . prim1 x2 x0 ⟶ x1 x2 ⟶ prim1 x2 (1216a.. x0 x1) (proof)
Theorem 6982e.. : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . prim1 x2 (1216a.. x0 x1) ⟶ and (prim1 x2 x0) (x1 x2) (proof)
Theorem 492ff.. : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . prim1 x2 (1216a.. x0 x1) ⟶ ∀ x3 : ο . (prim1 x2 x0 ⟶ x1 x2 ⟶ x3) ⟶ x3 (proof)
Definition a4c2a.. := λ x0 . λ x1 : ι → ο . 94f9e.. (1216a.. x0 x1)
Theorem e951d.. : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 : ι → ι . ∀ x3 . prim1 x3 x0 ⟶ x1 x3 ⟶ prim1 (x2 x3) (a4c2a.. x0 x1 x2) (proof)
Theorem ca584.. : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 : ι → ι . ∀ x3 . prim1 x3 (a4c2a.. x0 x1 x2) ⟶ ∀ x4 : ο . (∀ x5 . and (prim1 x5 x0) (and (x1 x5) (x3 = x2 x5)) ⟶ x4) ⟶ x4 (proof)
Theorem 932b3.. : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 : ι → ι . ∀ x3 . prim1 x3 (a4c2a.. x0 x1 x2) ⟶ ∀ x4 : ο . (∀ x5 . prim1 x5 x0 ⟶ x1 x5 ⟶ x3 = x2 x5 ⟶ x4) ⟶ x4 (proof)
Definition 3b429.. := λ x0 . λ x1 : ι → ι . λ x2 : ι → ι → ο . λ x3 : ι → ι → ι . prim3 (94f9e.. x0 (λ x4 . a4c2a.. (x1 x4) (x2 x4) (x3 x4)))
Theorem 81c0c.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ι → ι . ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 (x1 x4) ⟶ x2 x4 x5 ⟶ prim1 (x3 x4 x5) (3b429.. x0 x1 x2 x3) (proof)
Theorem 0cbcb.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ι → ι . ∀ x4 . prim1 x4 (3b429.. x0 x1 x2 x3) ⟶ ∀ x5 : ο . (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 (x1 x6) ⟶ x2 x6 x7 ⟶ x4 = x3 x6 x7 ⟶ x5) ⟶ x5 (proof)
Theorem c4419.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ι → ι . ∀ x4 . prim1 x4 (3b429.. x0 x1 x2 x3) ⟶ ∀ x5 : ο . (∀ x6 . and (prim1 x6 x0) (∀ x7 : ο . (∀ x8 . and (prim1 x8 (x1 x6)) (and (x2 x6 x8) (x4 = x3 x6 x8)) ⟶ x7) ⟶ x7) ⟶ x5) ⟶ x5 (proof)
Definition 85402.. := λ x0 . λ x1 : ι → ι . λ x2 : ι → ι → ι . λ x3 : ι → ι → ι → ο . λ x4 : ι → ι → ι → ι . prim3 (94f9e.. x0 (λ x5 . 3b429.. (x1 x5) (x2 x5) (x3 x5) (x4 x5)))
Theorem abff8.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ι → ο . ∀ x4 : ι → ι → ι → ι . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 (x1 x5) ⟶ ∀ x7 . prim1 x7 (x2 x5 x6) ⟶ x3 x5 x6 x7 ⟶ prim1 (x4 x5 x6 x7) (85402.. x0 x1 x2 x3 x4) (proof)
Theorem 9b3dc.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ι → ο . ∀ x4 : ι → ι → ι → ι . ∀ x5 . prim1 x5 (85402.. x0 x1 x2 x3 x4) ⟶ ∀ x6 : ο . (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 (x1 x7) ⟶ ∀ x9 . prim1 x9 (x2 x7 x8) ⟶ x3 x7 x8 x9 ⟶ x5 = x4 x7 x8 x9 ⟶ x6) ⟶ x6 (proof)
Theorem 7d382.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ι → ο . ∀ x4 : ι → ι → ι → ι . ∀ x5 . prim1 x5 (85402.. x0 x1 x2 x3 x4) ⟶ ∀ x6 : ο . (∀ x7 . and (prim1 x7 x0) (∀ x8 : ο . (∀ x9 . and (prim1 x9 (x1 x7)) (∀ x10 : ο . (∀ x11 . and (prim1 x11 (x2 x7 x9)) (and (x3 x7 x9 x11) (x5 = x4 x7 x9 x11)) ⟶ x10) ⟶ x10) ⟶ x8) ⟶ x8) ⟶ x6) ⟶ x6 (proof)
Definition f6efa.. := λ x0 . λ x1 : ι → ι . λ x2 : ι → ι → ι . λ x3 : ι → ι → ι → ι . λ x4 : ι → ι → ι → ι → ο . λ x5 : ι → ι → ι → ι → ι . prim3 (94f9e.. x0 (λ x6 . 85402.. (x1 x6) (x2 x6) (x3 x6) (x4 x6) (x5 x6)))
Theorem 05e93.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ι → ι . ∀ x4 : ι → ι → ι → ι → ο . ∀ x5 : ι → ι → ι → ι → ι . ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 (x1 x6) ⟶ ∀ x8 . prim1 x8 (x2 x6 x7) ⟶ ∀ x9 . prim1 x9 (x3 x6 x7 x8) ⟶ x4 x6 x7 x8 x9 ⟶ prim1 (x5 x6 x7 x8 x9) (f6efa.. x0 x1 x2 x3 x4 x5) (proof)
Theorem fd818.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ι → ι . ∀ x4 : ι → ι → ι → ι → ο . ∀ x5 : ι → ι → ι → ι → ι . ∀ x6 . prim1 x6 (f6efa.. x0 x1 x2 x3 x4 x5) ⟶ ∀ x7 : ο . (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 (x1 x8) ⟶ ∀ x10 . prim1 x10 (x2 x8 x9) ⟶ ∀ x11 . prim1 x11 (x3 x8 x9 x10) ⟶ x4 x8 x9 x10 x11 ⟶ x6 = x5 x8 x9 x10 x11 ⟶ x7) ⟶ x7 (proof)
Theorem 5a2be.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ι → ι . ∀ x4 : ι → ι → ι → ι → ο . ∀ x5 : ι → ι → ι → ι → ι . ∀ x6 . prim1 x6 (f6efa.. x0 x1 x2 x3 x4 x5) ⟶ ∀ x7 : ο . (∀ x8 . and (prim1 x8 x0) (∀ x9 : ο . (∀ x10 . and (prim1 x10 (x1 x8)) (∀ x11 : ο . (∀ x12 . and (prim1 x12 (x2 x8 x10)) (∀ x13 : ο . (∀ x14 . and (prim1 x14 (x3 x8 x10 x12)) (and (x4 x8 x10 x12 x14) (x6 = x5 x8 x10 x12 x14)) ⟶ x13) ⟶ x13) ⟶ x11) ⟶ x11) ⟶ x9) ⟶ x9) ⟶ x7) ⟶ x7 (proof)
Definition 2aab0.. := λ x0 . λ x1 : ι → ι . λ x2 : ι → ι → ι . λ x3 : ι → ι → ι → ι . λ x4 : ι → ι → ι → ι → ι . λ x5 : ι → ι → ι → ι → ι → ο . λ x6 : ι → ι → ι → ι → ι → ι . prim3 (94f9e.. x0 (λ x7 . f6efa.. (x1 x7) (x2 x7) (x3 x7) (x4 x7) (x5 x7) (x6 x7)))
Theorem d73e9.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ι → ι . ∀ x4 : ι → ι → ι → ι → ι . ∀ x5 : ι → ι → ι → ι → ι → ο . ∀ x6 : ι → ι → ι → ι → ι → ι . ∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 (x1 x7) ⟶ ∀ x9 . prim1 x9 (x2 x7 x8) ⟶ ∀ x10 . prim1 x10 (x3 x7 x8 x9) ⟶ ∀ x11 . prim1 x11 (x4 x7 x8 x9 x10) ⟶ x5 x7 x8 x9 x10 x11 ⟶ prim1 (x6 x7 x8 x9 x10 x11) (2aab0.. x0 x1 x2 x3 x4 x5 x6) (proof)
Theorem 7f213.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ι → ι . ∀ x4 : ι → ι → ι → ι → ι . ∀ x5 : ι → ι → ι → ι → ι → ο . ∀ x6 : ι → ι → ι → ι → ι → ι . ∀ x7 . prim1 x7 (2aab0.. x0 x1 x2 x3 x4 x5 x6) ⟶ ∀ x8 : ο . (∀ x9 . prim1 x9 x0 ⟶ ∀ x10 . prim1 x10 (x1 x9) ⟶ ∀ x11 . prim1 x11 (x2 x9 x10) ⟶ ∀ x12 . prim1 x12 (x3 x9 x10 x11) ⟶ ∀ x13 . prim1 x13 (x4 x9 x10 x11 x12) ⟶ x5 x9 x10 x11 x12 x13 ⟶ x7 = x6 x9 x10 x11 x12 x13 ⟶ x8) ⟶ x8 (proof)
Theorem 2e748.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ι → ι . ∀ x4 : ι → ι → ι → ι → ι . ∀ x5 : ι → ι → ι → ι → ι → ο . ∀ x6 : ι → ι → ι → ι → ι → ι . ∀ x7 . prim1 x7 (2aab0.. x0 x1 x2 x3 x4 x5 x6) ⟶ ∀ x8 : ο . (∀ x9 . and (prim1 x9 x0) (∀ x10 : ο . (∀ x11 . and (prim1 x11 (x1 x9)) (∀ x12 : ο . (∀ x13 . and (prim1 x13 (x2 x9 x11)) (∀ x14 : ο . (∀ x15 . and (prim1 x15 (x3 x9 x11 x13)) (∀ x16 : ο . (∀ x17 . and (prim1 x17 (x4 x9 x11 x13 x15)) (and (x5 x9 x11 x13 x15 x17) (x7 = x6 x9 x11 x13 x15 x17)) ⟶ x16) ⟶ x16) ⟶ x14) ⟶ x14) ⟶ x12) ⟶ x12) ⟶ x10) ⟶ x10) ⟶ x8) ⟶ x8 (proof)
Definition 6cd44.. := λ x0 . λ x1 : ι → ι . λ x2 : ι → ι → ι . λ x3 : ι → ι → ι → ι . λ x4 : ι → ι → ι → ι → ι . λ x5 : ι → ι → ι → ι → ι → ι . λ x6 : ι → ι → ι → ι → ι → ι → ο . λ x7 : ι → ι → ι → ι → ι → ι → ι . prim3 (94f9e.. x0 (λ x8 . 2aab0.. (x1 x8) (x2 x8) (x3 x8) (x4 x8) (x5 x8) (x6 x8) (x7 x8)))
Theorem 803ff.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ι → ι . ∀ x4 : ι → ι → ι → ι → ι . ∀ x5 : ι → ι → ι → ι → ι → ι . ∀ x6 : ι → ι → ι → ι → ι → ι → ο . ∀ x7 : ι → ι → ι → ι → ι → ι → ι . ∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 (x1 x8) ⟶ ∀ x10 . prim1 x10 (x2 x8 x9) ⟶ ∀ x11 . prim1 x11 (x3 x8 x9 x10) ⟶ ∀ x12 . prim1 x12 (x4 x8 x9 x10 x11) ⟶ ∀ x13 . prim1 x13 (x5 x8 x9 x10 x11 x12) ⟶ x6 x8 x9 x10 x11 x12 x13 ⟶ prim1 (x7 x8 x9 x10 x11 x12 x13) (6cd44.. x0 x1 x2 x3 x4 x5 x6 x7) (proof)
Theorem 8029f.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ι → ι . ∀ x4 : ι → ι → ι → ι → ι . ∀ x5 : ι → ι → ι → ι → ι → ι . ∀ x6 : ι → ι → ι → ι → ι → ι → ο . ∀ x7 : ι → ι → ι → ι → ι → ι → ι . ∀ x8 . prim1 x8 (6cd44.. x0 x1 x2 x3 x4 x5 x6 x7) ⟶ ∀ x9 : ο . (∀ x10 . prim1 x10 x0 ⟶ ∀ x11 . prim1 x11 (x1 x10) ⟶ ∀ x12 . prim1 x12 (x2 x10 x11) ⟶ ∀ x13 . prim1 x13 (x3 x10 x11 x12) ⟶ ∀ x14 . prim1 x14 (x4 x10 x11 x12 x13) ⟶ ∀ x15 . prim1 x15 (x5 x10 x11 x12 x13 x14) ⟶ x6 x10 x11 x12 x13 x14 x15 ⟶ x8 = x7 x10 x11 x12 x13 x14 x15 ⟶ x9) ⟶ x9 (proof)
Theorem 2ffba.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ι → ι . ∀ x4 : ι → ι → ι → ι → ι . ∀ x5 : ι → ι → ι → ι → ι → ι . ∀ x6 : ι → ι → ι → ι → ι → ι → ο . ∀ x7 : ι → ι → ι → ι → ι → ι → ι . ∀ x8 . prim1 x8 (6cd44.. x0 x1 x2 x3 x4 x5 x6 x7) ⟶ ∀ x9 : ο . (∀ x10 . and (prim1 x10 x0) (∀ x11 : ο . (∀ x12 . and (prim1 x12 (x1 x10)) (∀ x13 : ο . (∀ x14 . and (prim1 x14 (x2 x10 x12)) (∀ x15 : ο . (∀ x16 . and (prim1 x16 (x3 x10 x12 x14)) (∀ x17 : ο . (∀ x18 . and (prim1 x18 (x4 x10 x12 x14 x16)) (∀ x19 : ο . (∀ x20 . and (prim1 x20 (x5 x10 x12 x14 x16 x18)) (and (x6 x10 x12 x14 x16 x18 x20) (x8 = x7 x10 x12 x14 x16 x18 x20)) ⟶ x19) ⟶ x19) ⟶ x17) ⟶ x17) ⟶ x15) ⟶ x15) ⟶ x13) ⟶ x13) ⟶ x11) ⟶ x11) ⟶ x9) ⟶ x9 (proof)
Definition e5d4c.. := λ x0 . λ x1 : ι → ι . λ x2 : ι → ι → ι . λ x3 : ι → ι → ι → ι . λ x4 : ι → ι → ι → ι → ι . λ x5 : ι → ι → ι → ι → ι → ι . λ x6 : ι → ι → ι → ι → ι → ι → ι . λ x7 : ι → ι → ι → ι → ι → ι → ι → ο . λ x8 : ι → ι → ι → ι → ι → ι → ι → ι . prim3 (94f9e.. x0 (λ x9 . 6cd44.. (x1 x9) (x2 x9) (x3 x9) (x4 x9) (x5 x9) (x6 x9) (x7 x9) (x8 x9)))
Theorem 79851.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ι → ι . ∀ x4 : ι → ι → ι → ι → ι . ∀ x5 : ι → ι → ι → ι → ι → ι . ∀ x6 : ι → ι → ι → ι → ι → ι → ι . ∀ x7 : ι → ι → ι → ι → ι → ι → ι → ο . ∀ x8 : ι → ι → ι → ι → ι → ι → ι → ι . ∀ x9 . prim1 x9 x0 ⟶ ∀ x10 . prim1 x10 (x1 x9) ⟶ ∀ x11 . prim1 x11 (x2 x9 x10) ⟶ ∀ x12 . prim1 x12 (x3 x9 x10 x11) ⟶ ∀ x13 . prim1 x13 (x4 x9 x10 x11 x12) ⟶ ∀ x14 . prim1 x14 (x5 x9 x10 x11 x12 x13) ⟶ ∀ x15 . prim1 x15 (x6 x9 x10 x11 x12 x13 x14) ⟶ x7 x9 x10 x11 x12 x13 x14 x15 ⟶ prim1 (x8 x9 x10 x11 x12 x13 x14 x15) (e5d4c.. x0 x1 x2 x3 x4 x5 x6 x7 x8) (proof)
Theorem 48be8.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ι → ι . ∀ x4 : ι → ι → ι → ι → ι . ∀ x5 : ι → ι → ι → ι → ι → ι . ∀ x6 : ι → ι → ι → ι → ι → ι → ι . ∀ x7 : ι → ι → ι → ι → ι → ι → ι → ο . ∀ x8 : ι → ι → ι → ι → ι → ι → ι → ι . ∀ x9 . prim1 x9 (e5d4c.. x0 x1 x2 x3 x4 x5 x6 x7 x8) ⟶ ∀ x10 : ο . (∀ x11 . prim1 x11 x0 ⟶ ∀ x12 . prim1 x12 (x1 x11) ⟶ ∀ x13 . prim1 x13 (x2 x11 x12) ⟶ ∀ x14 . prim1 x14 (x3 x11 x12 x13) ⟶ ∀ x15 . prim1 x15 (x4 x11 x12 x13 x14) ⟶ ∀ x16 . prim1 x16 (x5 x11 x12 x13 x14 x15) ⟶ ∀ x17 . prim1 x17 (x6 x11 x12 x13 x14 x15 x16) ⟶ x7 x11 x12 x13 x14 x15 x16 x17 ⟶ x9 = x8 x11 x12 x13 x14 x15 x16 x17 ⟶ x10) ⟶ x10 (proof)
Theorem fb4aa.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ι → ι . ∀ x4 : ι → ι → ι → ι → ι . ∀ x5 : ι → ι → ι → ι → ι → ι . ∀ x6 : ι → ι → ι → ι → ι → ι → ι . ∀ x7 : ι → ι → ι → ι → ι → ι → ι → ο . ∀ x8 : ι → ι → ι → ι → ι → ι → ι → ι . ∀ x9 . prim1 x9 (e5d4c.. x0 x1 x2 x3 x4 x5 x6 x7 x8) ⟶ ∀ x10 : ο . (∀ x11 . and (prim1 x11 x0) (∀ x12 : ο . (∀ x13 . and (prim1 x13 (x1 x11)) (∀ x14 : ο . (∀ x15 . and (prim1 x15 (x2 x11 x13)) (∀ x16 : ο . (∀ x17 . and (prim1 x17 (x3 x11 x13 x15)) (∀ x18 : ο . (∀ x19 . and (prim1 x19 (x4 x11 x13 x15 x17)) (∀ x20 : ο . (∀ x21 . and (prim1 x21 (x5 x11 x13 x15 x17 x19)) (∀ x22 : ο . (∀ x23 . and (prim1 x23 (x6 x11 x13 x15 x17 x19 x21)) (and (x7 x11 x13 x15 x17 x19 x21 x23) (x9 = x8 x11 x13 x15 x17 x19 x21 x23)) ⟶ x22) ⟶ x22) ⟶ x20) ⟶ x20) ⟶ x18) ⟶ x18) ⟶ x16) ⟶ x16) ⟶ x14) ⟶ x14) ⟶ x12) ⟶ x12) ⟶ x10) ⟶ x10 (proof)
Definition f8922.. := λ x0 : (ι → ι) → ο . ∀ x1 : (ι → ι) → ι . ∀ x2 : ο . (∀ x3 . (∀ x4 : ι → ι . x0 x4 ⟶ x4 x3 = x1 x4) ⟶ x2) ⟶ x2
Theorem 674bb.. : ∀ x0 x1 : (ι → ι) → ο . f8922.. x1 ⟶ (∀ x2 : ι → ι . x0 x2 ⟶ x1 x2) ⟶ f8922.. x0 (proof)
Definition fa4ab.. := λ x0 : (ι → ι) → ο . λ x1 : (ι → ι) → ι . prim0 (λ x2 . ∀ x3 : ι → ι . x0 x3 ⟶ x3 x2 = x1 x3)
Theorem 0f560.. : ∀ x0 : (ι → ι) → ο . f8922.. x0 ⟶ ∀ x1 : ι → ι . x0 x1 ⟶ ∀ x2 : (ι → ι) → ι . x1 (fa4ab.. x0 x2) = x2 x1 (proof)
Definition e8533.. := λ x0 : (ι → ι) → ο . λ x1 . x1 = fa4ab.. x0 (λ x3 : ι → ι . x3 x1)
Theorem a4074.. : ∀ x0 : (ι → ι) → ο . f8922.. x0 ⟶ ∀ x1 . fa4ab.. x0 (λ x3 : ι → ι . x3 x1) = fa4ab.. x0 (λ x3 : ι → ι . x3 (fa4ab.. x0 (λ x4 : ι → ι . x4 x1))) (proof)
Theorem 5bcd7.. : ∀ x0 : (ι → ι) → ο . f8922.. x0 ⟶ ∀ x1 . e8533.. x0 (fa4ab.. x0 (λ x2 : ι → ι . x2 x1)) (proof)
Definition cdba9.. := λ x0 : (ι → ι) → ο . λ x1 x2 . ∀ x3 : ι → ι . x0 x3 ⟶ x3 x1 = x3 x2
Theorem 3c94a.. : ∀ x0 : (ι → ι) → ο . ∀ x1 . cdba9.. x0 x1 x1 (proof)
Theorem b8753.. : ∀ x0 : (ι → ι) → ο . ∀ x1 x2 . cdba9.. x0 x1 x2 ⟶ cdba9.. x0 x2 x1 (proof)
Theorem 43dff.. : ∀ x0 : (ι → ι) → ο . ∀ x1 x2 x3 . cdba9.. x0 x1 x2 ⟶ cdba9.. x0 x2 x3 ⟶ cdba9.. x0 x1 x3 (proof)
Definition 7bebd.. := λ x0 : ι → ι . λ x1 . True
Definition 5eb06.. := λ x0 : ι → ι . λ x1 . and (7bebd.. x0 x1) (e8533.. (λ x2 : ι → ι . x2 = x0) x1)
Definition cde91.. := λ x0 : ι → ι . λ x1 . ∀ x2 . nIn x2 (x0 x1)
Definition 52b09.. := λ x0 : ι → ι . λ x1 . not (cde91.. x0 x1)
Definition 9a7b4.. := λ x0 x1 : ι → ι . λ x2 . prim1 (x1 x2) (x0 x2)
Definition 1ba2d.. := λ x0 x1 : ι → ι . λ x2 . and (9a7b4.. x0 x1 x2) (e8533.. (λ x3 : ι → ι . or (x3 = x0) (x3 = x1)) x2)
Definition fa71f.. := λ x0 x1 x2 : ι → ι . λ x3 . and (9a7b4.. x0 x1 x3) (9a7b4.. x0 x2 x3)
Definition 3c39d.. := λ x0 x1 x2 : ι → ι . λ x3 . and (fa71f.. x0 x1 x2 x3) (e8533.. (λ x4 : ι → ι . or (or (x4 = x0) (x4 = x1)) (x4 = x2)) x3)
Theorem 500f6.. : ∀ x0 x1 : ι → ι . not (x0 = x1) ⟶ f8922.. (λ x2 : ι → ι . or (x2 = x0) (x2 = x1)) ⟶ ∀ x2 . 9a7b4.. x0 x1 x2 ⟶ 7bebd.. x0 x2 (proof)
Theorem 500f6.. : ∀ x0 x1 : ι → ι . not (x0 = x1) ⟶ f8922.. (λ x2 : ι → ι . or (x2 = x0) (x2 = x1)) ⟶ ∀ x2 . 9a7b4.. x0 x1 x2 ⟶ 7bebd.. x0 x2 (proof)
Theorem ea617.. : ∀ x0 x1 x2 : ι → ι . not (x0 = x1) ⟶ not (x0 = x2) ⟶ not (x1 = x2) ⟶ f8922.. (λ x3 : ι → ι . or (or (x3 = x0) (x3 = x1)) (x3 = x2)) ⟶ ∀ x3 . fa71f.. x0 x1 x2 x3 ⟶ 9a7b4.. x0 x1 x3 (proof)
Theorem e5856.. : ∀ x0 x1 x2 : ι → ι . not (x0 = x1) ⟶ not (x0 = x2) ⟶ not (x1 = x2) ⟶ f8922.. (λ x3 : ι → ι . or (or (x3 = x0) (x3 = x1)) (x3 = x2)) ⟶ ∀ x3 . fa71f.. x0 x1 x2 x3 ⟶ 9a7b4.. x0 x2 x3 (proof)
Definition 6ce64.. := λ x0 x1 : ι → ι . λ x2 . True
Definition 17176.. := λ x0 x1 : ι → ι . λ x2 . and (6ce64.. x0 x1 x2) (e8533.. (λ x3 : ι → ι . or (x3 = x0) (x3 = x1)) x2)