Proofgold Address

Param 0fc90.. : ι → (ι → ι) → ι
Param 4ae4a.. : ι → ι
Param 4a7ef.. : ι
Param If_i : ο → ι → ι → ι
Param e0e40.. : ι → ((ι → ο) → ο) → ι
Param d2155.. : ι → (ι → ι → ο) → ι
Param 1216a.. : ι → (ι → ο) → ι
Definition 7ac32.. := λ x0 . λ x1 : (ι → ο) → ο . λ x2 : ι → ι . λ x3 : ι → ι → ο . λ x4 : ι → ο . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (d2155.. x0 x3) (1216a.. x0 x4)))))
Param f482f.. : ι → ι → ι
Known 7d2e2.. : ∀ x0 x1 x2 x3 x4 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x6 . If_i (x6 = 4a7ef..) x0 (If_i (x6 = 4ae4a.. 4a7ef..) x1 (If_i (x6 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 (If_i (x6 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x3 x4))))) 4a7ef.. = x0
Theorem 8208e.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . x0 = 7ac32.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 4b222.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . x0 = f482f.. (7ac32.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Param decode_c : ι → (ι → ο) → ο
Known 504a8.. : ∀ x0 x1 x2 x3 x4 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x6 . If_i (x6 = 4a7ef..) x0 (If_i (x6 = 4ae4a.. 4a7ef..) x1 (If_i (x6 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 (If_i (x6 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x3 x4))))) (4ae4a.. 4a7ef..) = x1
Known 81500.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ο . (∀ x3 . x2 x3 ⟶ prim1 x3 x0) ⟶ decode_c (e0e40.. x0 x1) x2 = x1 x2
Theorem 9e1cf.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . x0 = 7ac32.. x1 x2 x3 x4 x5 ⟶ ∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x1) ⟶ x2 x6 = decode_c (f482f.. x0 (4ae4a.. 4a7ef..)) x6 (proof)
Theorem 84fcd.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 : ι → ο . (∀ x6 . x5 x6 ⟶ prim1 x6 x0) ⟶ x1 x5 = decode_c (f482f.. (7ac32.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 (proof)
Known fb20c.. : ∀ x0 x1 x2 x3 x4 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x6 . If_i (x6 = 4a7ef..) x0 (If_i (x6 = 4ae4a.. 4a7ef..) x1 (If_i (x6 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 (If_i (x6 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x3 x4))))) (4ae4a.. (4ae4a.. 4a7ef..)) = x2
Known f22ec.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 x0 ⟶ f482f.. (0fc90.. x0 x1) x2 = x1 x2
Theorem b554d.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . x0 = 7ac32.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x3 x6 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 (proof)
Theorem 41de0.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x2 x5 = f482f.. (f482f.. (7ac32.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Param 2b2e3.. : ι → ι → ι → ο
Known 431f3.. : ∀ x0 x1 x2 x3 x4 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x6 . If_i (x6 = 4a7ef..) x0 (If_i (x6 = 4ae4a.. 4a7ef..) x1 (If_i (x6 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 (If_i (x6 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x3 x4))))) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) = x3
Known 67416.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ 2b2e3.. (d2155.. x0 x1) x2 x3 = x1 x2 x3
Theorem cc9d8.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . x0 = 7ac32.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x4 x6 x7 = 2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 x7 (proof)
Theorem 627d4.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x3 x5 x6 = 2b2e3.. (f482f.. (7ac32.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 x6 (proof)
Param decode_p : ι → ι → ο
Known ffdcd.. : ∀ x0 x1 x2 x3 x4 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x6 . If_i (x6 = 4a7ef..) x0 (If_i (x6 = 4ae4a.. 4a7ef..) x1 (If_i (x6 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 (If_i (x6 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x3 x4))))) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) = x4
Known 931fe.. : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . prim1 x2 x0 ⟶ decode_p (1216a.. x0 x1) x2 = x1 x2
Theorem 87375.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . x0 = 7ac32.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x5 x6 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x6 (proof)
Theorem 3cd86.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x4 x5 = decode_p (f482f.. (7ac32.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x5 (proof)
Definition and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known and5I : ∀ x0 x1 x2 x3 x4 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ and (and (and (and x0 x1) x2) x3) x4
Theorem 0facc.. : ∀ x0 x1 . ∀ x2 x3 : (ι → ο) → ο . ∀ x4 x5 : ι → ι . ∀ x6 x7 : ι → ι → ο . ∀ x8 x9 : ι → ο . 7ac32.. x0 x2 x4 x6 x8 = 7ac32.. x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 : ι → ο . (∀ x11 . x10 x11 ⟶ prim1 x11 x0) ⟶ x2 x10 = x3 x10)) (∀ x10 . prim1 x10 x0 ⟶ x4 x10 = x5 x10)) (∀ x10 . prim1 x10 x0 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x6 x10 x11 = x7 x10 x11)) (∀ x10 . prim1 x10 x0 ⟶ x8 x10 = x9 x10) (proof)
Param iff : ο → ο → ο
Known ee7ef.. : ∀ x0 . ∀ x1 x2 : ι → ο . (∀ x3 . prim1 x3 x0 ⟶ iff (x1 x3) (x2 x3)) ⟶ 1216a.. x0 x1 = 1216a.. x0 x2
Known 62ef7.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ iff (x1 x3 x4) (x2 x3 x4)) ⟶ d2155.. x0 x1 = d2155.. x0 x2
Known 4402a.. : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ x1 x3 = x2 x3) ⟶ 0fc90.. x0 x1 = 0fc90.. x0 x2
Known fe043.. : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . (∀ x3 : ι → ο . (∀ x4 . x3 x4 ⟶ prim1 x4 x0) ⟶ iff (x1 x3) (x2 x3)) ⟶ e0e40.. x0 x1 = e0e40.. x0 x2
Theorem 924a0.. : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . ∀ x3 x4 : ι → ι . ∀ x5 x6 : ι → ι → ο . ∀ x7 x8 : ι → ο . (∀ x9 : ι → ο . (∀ x10 . x9 x10 ⟶ prim1 x10 x0) ⟶ iff (x1 x9) (x2 x9)) ⟶ (∀ x9 . prim1 x9 x0 ⟶ x3 x9 = x4 x9) ⟶ (∀ x9 . prim1 x9 x0 ⟶ ∀ x10 . prim1 x10 x0 ⟶ iff (x5 x9 x10) (x6 x9 x10)) ⟶ (∀ x9 . prim1 x9 x0 ⟶ iff (x7 x9) (x8 x9)) ⟶ 7ac32.. x0 x1 x3 x5 x7 = 7ac32.. x0 x2 x4 x6 x8 (proof)
Definition 687c3.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : (ι → ο) → ο . ∀ x4 : ι → ι . (∀ x5 . prim1 x5 x2 ⟶ prim1 (x4 x5) x2) ⟶ ∀ x5 : ι → ι → ο . ∀ x6 : ι → ο . x1 (7ac32.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem 8bc32.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) x0) ⟶ ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . 687c3.. (7ac32.. x0 x1 x2 x3 x4) (proof)
Theorem 6dc05.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . 687c3.. (7ac32.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x2 x5) x0 (proof)
Known iff_refl : ∀ x0 : ο . iff x0 x0
Theorem 3735e.. : ∀ x0 . 687c3.. x0 ⟶ x0 = 7ac32.. (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))) (proof)
Definition cf688.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → ι → ο) → (ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem a36e4.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → ι → ο) → (ι → ο) → ι . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x1) ⟶ iff (x2 x7) (x6 x7)) ⟶ ∀ x7 : ι → ι . (∀ x8 . prim1 x8 x1 ⟶ x3 x8 = x7 x8) ⟶ ∀ x8 : ι → ι → ο . (∀ x9 . prim1 x9 x1 ⟶ ∀ x10 . prim1 x10 x1 ⟶ iff (x4 x9 x10) (x8 x9 x10)) ⟶ ∀ x9 : ι → ο . (∀ x10 . prim1 x10 x1 ⟶ iff (x5 x10) (x9 x10)) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5) ⟶ cf688.. (7ac32.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 27e22.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → ι → ο) → (ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem 651a8.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → ι → ο) → (ι → ο) → ο . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x1) ⟶ iff (x2 x7) (x6 x7)) ⟶ ∀ x7 : ι → ι . (∀ x8 . prim1 x8 x1 ⟶ x3 x8 = x7 x8) ⟶ ∀ x8 : ι → ι → ο . (∀ x9 . prim1 x9 x1 ⟶ ∀ x10 . prim1 x10 x1 ⟶ iff (x4 x9 x10) (x8 x9 x10)) ⟶ ∀ x9 : ι → ο . (∀ x10 . prim1 x10 x1 ⟶ iff (x5 x10) (x9 x10)) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5) ⟶ 27e22.. (7ac32.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 608f4.. := λ x0 . λ x1 : (ι → ο) → ο . λ x2 : ι → ι . λ x3 : ι → ι → ο . λ x4 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (d2155.. x0 x3) x4))))
Theorem d8071.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . x0 = 608f4.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem aa832.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . x0 = f482f.. (608f4.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Theorem 60346.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . x0 = 608f4.. x1 x2 x3 x4 x5 ⟶ ∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x1) ⟶ x2 x6 = decode_c (f482f.. x0 (4ae4a.. 4a7ef..)) x6 (proof)
Theorem fd38c.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . ∀ x5 : ι → ο . (∀ x6 . x5 x6 ⟶ prim1 x6 x0) ⟶ x1 x5 = decode_c (f482f.. (608f4.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 (proof)
Theorem 97f65.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . x0 = 608f4.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x3 x6 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 (proof)
Theorem 1b0ab.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . prim1 x5 x0 ⟶ x2 x5 = f482f.. (f482f.. (608f4.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Theorem 1e09f.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . x0 = 608f4.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x4 x6 x7 = 2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 x7 (proof)
Theorem 24e08.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x3 x5 x6 = 2b2e3.. (f482f.. (608f4.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 x6 (proof)
Theorem c359b.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . x0 = 608f4.. x1 x2 x3 x4 x5 ⟶ x5 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem bf20a.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . x4 = f482f.. (608f4.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem 3ce02.. : ∀ x0 x1 . ∀ x2 x3 : (ι → ο) → ο . ∀ x4 x5 : ι → ι . ∀ x6 x7 : ι → ι → ο . ∀ x8 x9 . 608f4.. x0 x2 x4 x6 x8 = 608f4.. x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 : ι → ο . (∀ x11 . x10 x11 ⟶ prim1 x11 x0) ⟶ x2 x10 = x3 x10)) (∀ x10 . prim1 x10 x0 ⟶ x4 x10 = x5 x10)) (∀ x10 . prim1 x10 x0 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x6 x10 x11 = x7 x10 x11)) (x8 = x9) (proof)
Theorem dc4c2.. : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . ∀ x3 x4 : ι → ι . ∀ x5 x6 : ι → ι → ο . ∀ x7 . (∀ x8 : ι → ο . (∀ x9 . x8 x9 ⟶ prim1 x9 x0) ⟶ iff (x1 x8) (x2 x8)) ⟶ (∀ x8 . prim1 x8 x0 ⟶ x3 x8 = x4 x8) ⟶ (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ iff (x5 x8 x9) (x6 x8 x9)) ⟶ 608f4.. x0 x1 x3 x5 x7 = 608f4.. x0 x2 x4 x6 x7 (proof)
Definition d4af0.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : (ι → ο) → ο . ∀ x4 : ι → ι . (∀ x5 . prim1 x5 x2 ⟶ prim1 (x4 x5) x2) ⟶ ∀ x5 : ι → ι → ο . ∀ x6 . prim1 x6 x2 ⟶ x1 (608f4.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem e6954.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) x0) ⟶ ∀ x3 : ι → ι → ο . ∀ x4 . prim1 x4 x0 ⟶ d4af0.. (608f4.. x0 x1 x2 x3 x4) (proof)
Theorem 82387.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . d4af0.. (608f4.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x2 x5) x0 (proof)
Theorem e9780.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . d4af0.. (608f4.. x0 x1 x2 x3 x4) ⟶ prim1 x4 x0 (proof)
Theorem e13bf.. : ∀ x0 . d4af0.. x0 ⟶ x0 = 608f4.. (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition 3259d.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → ι → ο) → ι → ι . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 4f310.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → ι → ο) → ι → ι . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x1) ⟶ iff (x2 x7) (x6 x7)) ⟶ ∀ x7 : ι → ι . (∀ x8 . prim1 x8 x1 ⟶ x3 x8 = x7 x8) ⟶ ∀ x8 : ι → ι → ο . (∀ x9 . prim1 x9 x1 ⟶ ∀ x10 . prim1 x10 x1 ⟶ iff (x4 x9 x10) (x8 x9 x10)) ⟶ x0 x1 x6 x7 x8 x5 = x0 x1 x2 x3 x4 x5) ⟶ 3259d.. (608f4.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 85724.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → ι → ο) → ι → ο . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 1c3a3.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → ι → ο) → ι → ο . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x1) ⟶ iff (x2 x7) (x6 x7)) ⟶ ∀ x7 : ι → ι . (∀ x8 . prim1 x8 x1 ⟶ x3 x8 = x7 x8) ⟶ ∀ x8 : ι → ι → ο . (∀ x9 . prim1 x9 x1 ⟶ ∀ x10 . prim1 x10 x1 ⟶ iff (x4 x9 x10) (x8 x9 x10)) ⟶ x0 x1 x6 x7 x8 x5 = x0 x1 x2 x3 x4 x5) ⟶ 85724.. (608f4.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition f4fb4.. := λ x0 . λ x1 : (ι → ο) → ο . λ x2 : ι → ι . λ x3 x4 : ι → ο . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (1216a.. x0 x3) (1216a.. x0 x4)))))
Theorem 7c086.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 x5 : ι → ο . x0 = f4fb4.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem a0755.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 x4 : ι → ο . x0 = f482f.. (f4fb4.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Theorem 9e061.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 x5 : ι → ο . x0 = f4fb4.. x1 x2 x3 x4 x5 ⟶ ∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x1) ⟶ x2 x6 = decode_c (f482f.. x0 (4ae4a.. 4a7ef..)) x6 (proof)
Theorem 5fca6.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 x4 x5 : ι → ο . (∀ x6 . x5 x6 ⟶ prim1 x6 x0) ⟶ x1 x5 = decode_c (f482f.. (f4fb4.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 (proof)
Theorem 5ca46.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 x5 : ι → ο . x0 = f4fb4.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x3 x6 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 (proof)
Theorem 95606.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x2 x5 = f482f.. (f482f.. (f4fb4.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Theorem a66f2.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 x5 : ι → ο . x0 = f4fb4.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x4 x6 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 (proof)
Theorem c4d00.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x3 x5 = decode_p (f482f.. (f4fb4.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 (proof)
Theorem 298cb.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 x5 : ι → ο . x0 = f4fb4.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x5 x6 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x6 (proof)
Theorem 58f90.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x4 x5 = decode_p (f482f.. (f4fb4.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x5 (proof)
Theorem c1ea6.. : ∀ x0 x1 . ∀ x2 x3 : (ι → ο) → ο . ∀ x4 x5 : ι → ι . ∀ x6 x7 x8 x9 : ι → ο . f4fb4.. x0 x2 x4 x6 x8 = f4fb4.. x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 : ι → ο . (∀ x11 . x10 x11 ⟶ prim1 x11 x0) ⟶ x2 x10 = x3 x10)) (∀ x10 . prim1 x10 x0 ⟶ x4 x10 = x5 x10)) (∀ x10 . prim1 x10 x0 ⟶ x6 x10 = x7 x10)) (∀ x10 . prim1 x10 x0 ⟶ x8 x10 = x9 x10) (proof)
Theorem f74bd.. : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . ∀ x3 x4 : ι → ι . ∀ x5 x6 x7 x8 : ι → ο . (∀ x9 : ι → ο . (∀ x10 . x9 x10 ⟶ prim1 x10 x0) ⟶ iff (x1 x9) (x2 x9)) ⟶ (∀ x9 . prim1 x9 x0 ⟶ x3 x9 = x4 x9) ⟶ (∀ x9 . prim1 x9 x0 ⟶ iff (x5 x9) (x6 x9)) ⟶ (∀ x9 . prim1 x9 x0 ⟶ iff (x7 x9) (x8 x9)) ⟶ f4fb4.. x0 x1 x3 x5 x7 = f4fb4.. x0 x2 x4 x6 x8 (proof)
Definition 24ba9.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : (ι → ο) → ο . ∀ x4 : ι → ι . (∀ x5 . prim1 x5 x2 ⟶ prim1 (x4 x5) x2) ⟶ ∀ x5 x6 : ι → ο . x1 (f4fb4.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem 70513.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) x0) ⟶ ∀ x3 x4 : ι → ο . 24ba9.. (f4fb4.. x0 x1 x2 x3 x4) (proof)
Theorem 819e3.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 x4 : ι → ο . 24ba9.. (f4fb4.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x2 x5) x0 (proof)
Theorem b1dc2.. : ∀ x0 . 24ba9.. x0 ⟶ x0 = f4fb4.. (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))) (proof)
Definition 16885.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → ο) → (ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem 63a8c.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → ο) → (ι → ο) → ι . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 x5 : ι → ο . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x1) ⟶ iff (x2 x7) (x6 x7)) ⟶ ∀ x7 : ι → ι . (∀ x8 . prim1 x8 x1 ⟶ x3 x8 = x7 x8) ⟶ ∀ x8 : ι → ο . (∀ x9 . prim1 x9 x1 ⟶ iff (x4 x9) (x8 x9)) ⟶ ∀ x9 : ι → ο . (∀ x10 . prim1 x10 x1 ⟶ iff (x5 x10) (x9 x10)) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5) ⟶ 16885.. (f4fb4.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 08fa6.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → ο) → (ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem be7e4.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → ο) → (ι → ο) → ο . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 x5 : ι → ο . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x1) ⟶ iff (x2 x7) (x6 x7)) ⟶ ∀ x7 : ι → ι . (∀ x8 . prim1 x8 x1 ⟶ x3 x8 = x7 x8) ⟶ ∀ x8 : ι → ο . (∀ x9 . prim1 x9 x1 ⟶ iff (x4 x9) (x8 x9)) ⟶ ∀ x9 : ι → ο . (∀ x10 . prim1 x10 x1 ⟶ iff (x5 x10) (x9 x10)) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5) ⟶ 08fa6.. (f4fb4.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 2e40c.. := λ x0 . λ x1 : (ι → ο) → ο . λ x2 : ι → ι . λ x3 : ι → ο . λ x4 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (1216a.. x0 x3) x4))))
Theorem acfd5.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ο . ∀ x5 . x0 = 2e40c.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem f77a7.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ο . ∀ x4 . x0 = f482f.. (2e40c.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Theorem 7075a.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ο . ∀ x5 . x0 = 2e40c.. x1 x2 x3 x4 x5 ⟶ ∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x1) ⟶ x2 x6 = decode_c (f482f.. x0 (4ae4a.. 4a7ef..)) x6 (proof)
Theorem 071f5.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ο . ∀ x4 . ∀ x5 : ι → ο . (∀ x6 . x5 x6 ⟶ prim1 x6 x0) ⟶ x1 x5 = decode_c (f482f.. (2e40c.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 (proof)
Theorem dbbce.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ο . ∀ x5 . x0 = 2e40c.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x3 x6 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 (proof)
Theorem dc308.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ο . ∀ x4 x5 . prim1 x5 x0 ⟶ x2 x5 = f482f.. (f482f.. (2e40c.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Theorem 528c2.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ο . ∀ x5 . x0 = 2e40c.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x4 x6 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 (proof)
Theorem 9b6e5.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ο . ∀ x4 x5 . prim1 x5 x0 ⟶ x3 x5 = decode_p (f482f.. (2e40c.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 (proof)
Theorem 846b3.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ο . ∀ x5 . x0 = 2e40c.. x1 x2 x3 x4 x5 ⟶ x5 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem 67814.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ο . ∀ x4 . x4 = f482f.. (2e40c.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem 0f819.. : ∀ x0 x1 . ∀ x2 x3 : (ι → ο) → ο . ∀ x4 x5 : ι → ι . ∀ x6 x7 : ι → ο . ∀ x8 x9 . 2e40c.. x0 x2 x4 x6 x8 = 2e40c.. x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 : ι → ο . (∀ x11 . x10 x11 ⟶ prim1 x11 x0) ⟶ x2 x10 = x3 x10)) (∀ x10 . prim1 x10 x0 ⟶ x4 x10 = x5 x10)) (∀ x10 . prim1 x10 x0 ⟶ x6 x10 = x7 x10)) (x8 = x9) (proof)
Theorem 02f07.. : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . ∀ x3 x4 : ι → ι . ∀ x5 x6 : ι → ο . ∀ x7 . (∀ x8 : ι → ο . (∀ x9 . x8 x9 ⟶ prim1 x9 x0) ⟶ iff (x1 x8) (x2 x8)) ⟶ (∀ x8 . prim1 x8 x0 ⟶ x3 x8 = x4 x8) ⟶ (∀ x8 . prim1 x8 x0 ⟶ iff (x5 x8) (x6 x8)) ⟶ 2e40c.. x0 x1 x3 x5 x7 = 2e40c.. x0 x2 x4 x6 x7 (proof)
Definition 0670d.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : (ι → ο) → ο . ∀ x4 : ι → ι . (∀ x5 . prim1 x5 x2 ⟶ prim1 (x4 x5) x2) ⟶ ∀ x5 : ι → ο . ∀ x6 . prim1 x6 x2 ⟶ x1 (2e40c.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem c905c.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) x0) ⟶ ∀ x3 : ι → ο . ∀ x4 . prim1 x4 x0 ⟶ 0670d.. (2e40c.. x0 x1 x2 x3 x4) (proof)
Theorem 5a473.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ο . ∀ x4 . 0670d.. (2e40c.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x2 x5) x0 (proof)
Theorem 59c4d.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ο . ∀ x4 . 0670d.. (2e40c.. x0 x1 x2 x3 x4) ⟶ prim1 x4 x0 (proof)
Theorem 1c1fb.. : ∀ x0 . 0670d.. x0 ⟶ x0 = 2e40c.. (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition 2def8.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → ο) → ι → ι . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 7d0b3.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → ο) → ι → ι . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ο . ∀ x5 . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x1) ⟶ iff (x2 x7) (x6 x7)) ⟶ ∀ x7 : ι → ι . (∀ x8 . prim1 x8 x1 ⟶ x3 x8 = x7 x8) ⟶ ∀ x8 : ι → ο . (∀ x9 . prim1 x9 x1 ⟶ iff (x4 x9) (x8 x9)) ⟶ x0 x1 x6 x7 x8 x5 = x0 x1 x2 x3 x4 x5) ⟶ 2def8.. (2e40c.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 5592c.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → ο) → ι → ο . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 3db9b.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → ο) → ι → ο . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ο . ∀ x5 . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x1) ⟶ iff (x2 x7) (x6 x7)) ⟶ ∀ x7 : ι → ι . (∀ x8 . prim1 x8 x1 ⟶ x3 x8 = x7 x8) ⟶ ∀ x8 : ι → ο . (∀ x9 . prim1 x9 x1 ⟶ iff (x4 x9) (x8 x9)) ⟶ x0 x1 x6 x7 x8 x5 = x0 x1 x2 x3 x4 x5) ⟶ 5592c.. (2e40c.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)