Proofgold Asset

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
Definition 17e9f.. := λ x0 . λ x1 : (ι → ο) → ο . λ x2 : ι → ι . λ x3 : ι → ι → ο . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x4 . If_i (x4 = 4a7ef..) x0 (If_i (x4 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (If_i (x4 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. x0 x2) (d2155.. x0 x3))))
Param f482f.. : ι → ι → ι
Known 9f6be.. : ∀ x0 x1 x2 x3 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) x1 (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 x3)))) 4a7ef.. = x0
Theorem 7492d.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . x0 = 17e9f.. x1 x2 x3 x4 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem b56ab.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . x4 x0 (f482f.. (17e9f.. x0 x1 x2 x3) 4a7ef..) ⟶ x4 (f482f.. (17e9f.. x0 x1 x2 x3) 4a7ef..) x0 (proof)
Param decode_c : ι → (ι → ο) → ο
Known 8a328.. : ∀ x0 x1 x2 x3 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) x1 (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 x3)))) (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 691ba.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . x0 = 17e9f.. x1 x2 x3 x4 ⟶ ∀ x5 : ι → ο . (∀ x6 . x5 x6 ⟶ prim1 x6 x1) ⟶ x2 x5 = decode_c (f482f.. x0 (4ae4a.. 4a7ef..)) x5 (proof)
Theorem 155f9.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . (∀ x5 . x4 x5 ⟶ prim1 x5 x0) ⟶ x1 x4 = decode_c (f482f.. (17e9f.. x0 x1 x2 x3) (4ae4a.. 4a7ef..)) x4 (proof)
Known 142e6.. : ∀ x0 x1 x2 x3 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) x1 (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 x3)))) (4ae4a.. (4ae4a.. 4a7ef..)) = x2
Known f22ec.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 x0 ⟶ f482f.. (0fc90.. x0 x1) x2 = x1 x2
Theorem bcc00.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . x0 = 17e9f.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x3 x5 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Theorem b24dc.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . prim1 x4 x0 ⟶ x2 x4 = f482f.. (f482f.. (17e9f.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) x4 (proof)
Param 2b2e3.. : ι → ι → ι → ο
Known 62a6b.. : ∀ x0 x1 x2 x3 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) x1 (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 x3)))) (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 f28c1.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . x0 = 17e9f.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x4 x5 x6 = 2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 x6 (proof)
Theorem e1cfe.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x3 x4 x5 = 2b2e3.. (f482f.. (17e9f.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x4 x5 (proof)
Definition and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known and4I : ∀ x0 x1 x2 x3 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ and (and (and x0 x1) x2) x3
Theorem 4c1a8.. : ∀ x0 x1 . ∀ x2 x3 : (ι → ο) → ο . ∀ x4 x5 : ι → ι . ∀ x6 x7 : ι → ι → ο . 17e9f.. x0 x2 x4 x6 = 17e9f.. x1 x3 x5 x7 ⟶ and (and (and (x0 = x1) (∀ x8 : ι → ο . (∀ x9 . x8 x9 ⟶ prim1 x9 x0) ⟶ x2 x8 = x3 x8)) (∀ x8 . prim1 x8 x0 ⟶ x4 x8 = x5 x8)) (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x6 x8 x9 = x7 x8 x9) (proof)
Param iff : ο → ο → ο
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 44ab7.. : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . ∀ x3 x4 : ι → ι . ∀ x5 x6 : ι → ι → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x0) ⟶ iff (x1 x7) (x2 x7)) ⟶ (∀ x7 . prim1 x7 x0 ⟶ x3 x7 = x4 x7) ⟶ (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ iff (x5 x7 x8) (x6 x7 x8)) ⟶ 17e9f.. x0 x1 x3 x5 = 17e9f.. x0 x2 x4 x6 (proof)
Definition 9a300.. := 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 . x1 (17e9f.. x2 x3 x4 x5)) ⟶ x1 x0
Theorem 8a21b.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) x0) ⟶ ∀ x3 : ι → ι → ο . 9a300.. (17e9f.. x0 x1 x2 x3) (proof)
Theorem be260.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . 9a300.. (17e9f.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x4) x0 (proof)
Known iff_refl : ∀ x0 : ο . iff x0 x0
Theorem dcb35.. : ∀ x0 . 9a300.. x0 ⟶ x0 = 17e9f.. (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition 1fdeb.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → 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..)))))
Theorem 9565e.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → ι → ο) → ι . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . (∀ x5 : (ι → ο) → ο . (∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x1) ⟶ iff (x2 x6) (x5 x6)) ⟶ ∀ x6 : ι → ι . (∀ x7 . prim1 x7 x1 ⟶ x3 x7 = x6 x7) ⟶ ∀ x7 : ι → ι → ο . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ iff (x4 x8 x9) (x7 x8 x9)) ⟶ x0 x1 x5 x6 x7 = x0 x1 x2 x3 x4) ⟶ 1fdeb.. (17e9f.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition c1732.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → 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..)))))
Theorem d83ec.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → ι → ο) → ο . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . (∀ x5 : (ι → ο) → ο . (∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x1) ⟶ iff (x2 x6) (x5 x6)) ⟶ ∀ x6 : ι → ι . (∀ x7 . prim1 x7 x1 ⟶ x3 x7 = x6 x7) ⟶ ∀ x7 : ι → ι → ο . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ iff (x4 x8 x9) (x7 x8 x9)) ⟶ x0 x1 x5 x6 x7 = x0 x1 x2 x3 x4) ⟶ c1732.. (17e9f.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Param 1216a.. : ι → (ι → ο) → ι
Definition c7ccc.. := λ x0 . λ x1 : (ι → ο) → ο . λ x2 : ι → ι . λ x3 : ι → ο . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x4 . If_i (x4 = 4a7ef..) x0 (If_i (x4 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (If_i (x4 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. x0 x2) (1216a.. x0 x3))))
Theorem cec89.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ο . x0 = c7ccc.. x1 x2 x3 x4 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 36fd9.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ο . x0 = f482f.. (c7ccc.. x0 x1 x2 x3) 4a7ef.. (proof)
Theorem 47b2a.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ο . x0 = c7ccc.. x1 x2 x3 x4 ⟶ ∀ x5 : ι → ο . (∀ x6 . x5 x6 ⟶ prim1 x6 x1) ⟶ x2 x5 = decode_c (f482f.. x0 (4ae4a.. 4a7ef..)) x5 (proof)
Theorem 620b0.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 x4 : ι → ο . (∀ x5 . x4 x5 ⟶ prim1 x5 x0) ⟶ x1 x4 = decode_c (f482f.. (c7ccc.. x0 x1 x2 x3) (4ae4a.. 4a7ef..)) x4 (proof)
Theorem 9867d.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ο . x0 = c7ccc.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x3 x5 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Theorem 1da29.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ο . ∀ x4 . prim1 x4 x0 ⟶ x2 x4 = f482f.. (f482f.. (c7ccc.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) x4 (proof)
Param decode_p : ι → ι → ο
Known 931fe.. : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . prim1 x2 x0 ⟶ decode_p (1216a.. x0 x1) x2 = x1 x2
Theorem e5fea.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ο . x0 = c7ccc.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x4 x5 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 (proof)
Theorem 880ae.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ο . ∀ x4 . prim1 x4 x0 ⟶ x3 x4 = decode_p (f482f.. (c7ccc.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x4 (proof)
Theorem 9757d.. : ∀ x0 x1 . ∀ x2 x3 : (ι → ο) → ο . ∀ x4 x5 : ι → ι . ∀ x6 x7 : ι → ο . c7ccc.. x0 x2 x4 x6 = c7ccc.. x1 x3 x5 x7 ⟶ and (and (and (x0 = x1) (∀ x8 : ι → ο . (∀ x9 . x8 x9 ⟶ prim1 x9 x0) ⟶ x2 x8 = x3 x8)) (∀ x8 . prim1 x8 x0 ⟶ x4 x8 = x5 x8)) (∀ x8 . prim1 x8 x0 ⟶ x6 x8 = x7 x8) (proof)
Known ee7ef.. : ∀ x0 . ∀ x1 x2 : ι → ο . (∀ x3 . prim1 x3 x0 ⟶ iff (x1 x3) (x2 x3)) ⟶ 1216a.. x0 x1 = 1216a.. x0 x2
Theorem f4cca.. : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . ∀ x3 x4 : ι → ι . ∀ x5 x6 : ι → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x0) ⟶ iff (x1 x7) (x2 x7)) ⟶ (∀ x7 . prim1 x7 x0 ⟶ x3 x7 = x4 x7) ⟶ (∀ x7 . prim1 x7 x0 ⟶ iff (x5 x7) (x6 x7)) ⟶ c7ccc.. x0 x1 x3 x5 = c7ccc.. x0 x2 x4 x6 (proof)
Definition 49755.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : (ι → ο) → ο . ∀ x4 : ι → ι . (∀ x5 . prim1 x5 x2 ⟶ prim1 (x4 x5) x2) ⟶ ∀ x5 : ι → ο . x1 (c7ccc.. x2 x3 x4 x5)) ⟶ x1 x0
Theorem aa00f.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) x0) ⟶ ∀ x3 : ι → ο . 49755.. (c7ccc.. x0 x1 x2 x3) (proof)
Theorem d05db.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ο . 49755.. (c7ccc.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x4) x0 (proof)
Theorem 12038.. : ∀ x0 . 49755.. x0 ⟶ x0 = c7ccc.. (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition 13026.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι) → (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..)))))
Theorem f43cf.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → ο) → ι . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ο . (∀ x5 : (ι → ο) → ο . (∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x1) ⟶ iff (x2 x6) (x5 x6)) ⟶ ∀ x6 : ι → ι . (∀ x7 . prim1 x7 x1 ⟶ x3 x7 = x6 x7) ⟶ ∀ x7 : ι → ο . (∀ x8 . prim1 x8 x1 ⟶ iff (x4 x8) (x7 x8)) ⟶ x0 x1 x5 x6 x7 = x0 x1 x2 x3 x4) ⟶ 13026.. (c7ccc.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition 2fe05.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι) → (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..)))))
Theorem 8aa5c.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → ο) → ο . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 : ι → ο . (∀ x5 : (ι → ο) → ο . (∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x1) ⟶ iff (x2 x6) (x5 x6)) ⟶ ∀ x6 : ι → ι . (∀ x7 . prim1 x7 x1 ⟶ x3 x7 = x6 x7) ⟶ ∀ x7 : ι → ο . (∀ x8 . prim1 x8 x1 ⟶ iff (x4 x8) (x7 x8)) ⟶ x0 x1 x5 x6 x7 = x0 x1 x2 x3 x4) ⟶ 2fe05.. (c7ccc.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition eb2d9.. := λ x0 . λ x1 : (ι → ο) → ο . λ x2 : ι → ι . λ x3 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x4 . If_i (x4 = 4a7ef..) x0 (If_i (x4 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (If_i (x4 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. x0 x2) x3)))
Theorem 17a28.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 . x0 = eb2d9.. x1 x2 x3 x4 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 4b77a.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 . x0 = f482f.. (eb2d9.. x0 x1 x2 x3) 4a7ef.. (proof)
Theorem 9fde6.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 . x0 = eb2d9.. x1 x2 x3 x4 ⟶ ∀ x5 : ι → ο . (∀ x6 . x5 x6 ⟶ prim1 x6 x1) ⟶ x2 x5 = decode_c (f482f.. x0 (4ae4a.. 4a7ef..)) x5 (proof)
Theorem 0041c.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 . ∀ x4 : ι → ο . (∀ x5 . x4 x5 ⟶ prim1 x5 x0) ⟶ x1 x4 = decode_c (f482f.. (eb2d9.. x0 x1 x2 x3) (4ae4a.. 4a7ef..)) x4 (proof)
Theorem ae098.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 . x0 = eb2d9.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x3 x5 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Theorem 923f0.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 x4 . prim1 x4 x0 ⟶ x2 x4 = f482f.. (f482f.. (eb2d9.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) x4 (proof)
Theorem dc4aa.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 . x0 = eb2d9.. x1 x2 x3 x4 ⟶ x4 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem dc48a.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 . x3 = f482f.. (eb2d9.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem 7b502.. : ∀ x0 x1 . ∀ x2 x3 : (ι → ο) → ο . ∀ x4 x5 : ι → ι . ∀ x6 x7 . eb2d9.. x0 x2 x4 x6 = eb2d9.. x1 x3 x5 x7 ⟶ and (and (and (x0 = x1) (∀ x8 : ι → ο . (∀ x9 . x8 x9 ⟶ prim1 x9 x0) ⟶ x2 x8 = x3 x8)) (∀ x8 . prim1 x8 x0 ⟶ x4 x8 = x5 x8)) (x6 = x7) (proof)
Theorem f600a.. : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . ∀ x3 x4 : ι → ι . ∀ x5 . (∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x0) ⟶ iff (x1 x6) (x2 x6)) ⟶ (∀ x6 . prim1 x6 x0 ⟶ x3 x6 = x4 x6) ⟶ eb2d9.. x0 x1 x3 x5 = eb2d9.. x0 x2 x4 x5 (proof)
Definition b67a0.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : (ι → ο) → ο . ∀ x4 : ι → ι . (∀ x5 . prim1 x5 x2 ⟶ prim1 (x4 x5) x2) ⟶ ∀ x5 . prim1 x5 x2 ⟶ x1 (eb2d9.. x2 x3 x4 x5)) ⟶ x1 x0
Theorem 6b633.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) x0) ⟶ ∀ x3 . prim1 x3 x0 ⟶ b67a0.. (eb2d9.. x0 x1 x2 x3) (proof)
Theorem c0e69.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 . b67a0.. (eb2d9.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x4) x0 (proof)
Theorem 5c7ca.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 . b67a0.. (eb2d9.. x0 x1 x2 x3) ⟶ prim1 x3 x0 (proof)
Theorem 20b3f.. : ∀ x0 . b67a0.. x0 ⟶ x0 = eb2d9.. (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Definition 26d87.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι) → ι → ι . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem 9353f.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι) → ι → ι . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 . (∀ x5 : (ι → ο) → ο . (∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x1) ⟶ iff (x2 x6) (x5 x6)) ⟶ ∀ x6 : ι → ι . (∀ x7 . prim1 x7 x1 ⟶ x3 x7 = x6 x7) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4) ⟶ 26d87.. (eb2d9.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition 6c446.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι) → ι → ο . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem 2b25e.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι) → ι → ο . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 . (∀ x5 : (ι → ο) → ο . (∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x1) ⟶ iff (x2 x6) (x5 x6)) ⟶ ∀ x6 : ι → ι . (∀ x7 . prim1 x7 x1 ⟶ x3 x7 = x6 x7) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4) ⟶ 6c446.. (eb2d9.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition 30bff.. := λ x0 . λ x1 : (ι → ο) → ο . λ x2 : ι → ι → ο . λ x3 : ι → ο . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x4 . If_i (x4 = 4a7ef..) x0 (If_i (x4 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (If_i (x4 = 4ae4a.. (4ae4a.. 4a7ef..)) (d2155.. x0 x2) (1216a.. x0 x3))))
Theorem 6d1ab.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . x0 = 30bff.. x1 x2 x3 x4 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem a089c.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . x0 = f482f.. (30bff.. x0 x1 x2 x3) 4a7ef.. (proof)
Theorem 8da93.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . x0 = 30bff.. x1 x2 x3 x4 ⟶ ∀ x5 : ι → ο . (∀ x6 . x5 x6 ⟶ prim1 x6 x1) ⟶ x2 x5 = decode_c (f482f.. x0 (4ae4a.. 4a7ef..)) x5 (proof)
Theorem 66eb4.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ο . ∀ x3 x4 : ι → ο . (∀ x5 . x4 x5 ⟶ prim1 x5 x0) ⟶ x1 x4 = decode_c (f482f.. (30bff.. x0 x1 x2 x3) (4ae4a.. 4a7ef..)) x4 (proof)
Theorem b818c.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . x0 = 30bff.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x3 x5 x6 = 2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Theorem bd057.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x2 x4 x5 = 2b2e3.. (f482f.. (30bff.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) x4 x5 (proof)
Theorem 7b60c.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . x0 = 30bff.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x4 x5 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 (proof)
Theorem 3fc2d.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 . prim1 x4 x0 ⟶ x3 x4 = decode_p (f482f.. (30bff.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x4 (proof)
Theorem eabbd.. : ∀ x0 x1 . ∀ x2 x3 : (ι → ο) → ο . ∀ x4 x5 : ι → ι → ο . ∀ x6 x7 : ι → ο . 30bff.. x0 x2 x4 x6 = 30bff.. x1 x3 x5 x7 ⟶ and (and (and (x0 = x1) (∀ x8 : ι → ο . (∀ x9 . x8 x9 ⟶ prim1 x9 x0) ⟶ x2 x8 = x3 x8)) (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x4 x8 x9 = x5 x8 x9)) (∀ x8 . prim1 x8 x0 ⟶ x6 x8 = x7 x8) (proof)
Theorem b69b2.. : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . ∀ x3 x4 : ι → ι → ο . ∀ x5 x6 : ι → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x0) ⟶ iff (x1 x7) (x2 x7)) ⟶ (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ iff (x3 x7 x8) (x4 x7 x8)) ⟶ (∀ x7 . prim1 x7 x0 ⟶ iff (x5 x7) (x6 x7)) ⟶ 30bff.. x0 x1 x3 x5 = 30bff.. x0 x2 x4 x6 (proof)
Definition 014dd.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : (ι → ο) → ο . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . x1 (30bff.. x2 x3 x4 x5)) ⟶ x1 x0
Theorem b0689.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . 014dd.. (30bff.. x0 x1 x2 x3) (proof)
Theorem 2f936.. : ∀ x0 . 014dd.. x0 ⟶ x0 = 30bff.. (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition 7811e.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι → ο) → (ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 51deb.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι → ο) → (ι → ο) → ι . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . (∀ x5 : (ι → ο) → ο . (∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x1) ⟶ iff (x2 x6) (x5 x6)) ⟶ ∀ x6 : ι → ι → ο . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ iff (x3 x7 x8) (x6 x7 x8)) ⟶ ∀ x7 : ι → ο . (∀ x8 . prim1 x8 x1 ⟶ iff (x4 x8) (x7 x8)) ⟶ x0 x1 x5 x6 x7 = x0 x1 x2 x3 x4) ⟶ 7811e.. (30bff.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition f9779.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι → ο) → (ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 4755f.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι → ο) → (ι → ο) → ο . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . (∀ x5 : (ι → ο) → ο . (∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x1) ⟶ iff (x2 x6) (x5 x6)) ⟶ ∀ x6 : ι → ι → ο . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ iff (x3 x7 x8) (x6 x7 x8)) ⟶ ∀ x7 : ι → ο . (∀ x8 . prim1 x8 x1 ⟶ iff (x4 x8) (x7 x8)) ⟶ x0 x1 x5 x6 x7 = x0 x1 x2 x3 x4) ⟶ f9779.. (30bff.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition d8d01.. := λ x0 . λ x1 : (ι → ο) → ο . λ x2 : ι → ι → ο . λ x3 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x4 . If_i (x4 = 4a7ef..) x0 (If_i (x4 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (If_i (x4 = 4ae4a.. (4ae4a.. 4a7ef..)) (d2155.. x0 x2) x3)))
Theorem 61567.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ο . ∀ x4 . x0 = d8d01.. x1 x2 x3 x4 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 0d3cc.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ο . ∀ x3 . x0 = f482f.. (d8d01.. x0 x1 x2 x3) 4a7ef.. (proof)
Theorem 89e73.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ο . ∀ x4 . x0 = d8d01.. x1 x2 x3 x4 ⟶ ∀ x5 : ι → ο . (∀ x6 . x5 x6 ⟶ prim1 x6 x1) ⟶ x2 x5 = decode_c (f482f.. x0 (4ae4a.. 4a7ef..)) x5 (proof)
Theorem e80db.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ο . ∀ x3 . ∀ x4 : ι → ο . (∀ x5 . x4 x5 ⟶ prim1 x5 x0) ⟶ x1 x4 = decode_c (f482f.. (d8d01.. x0 x1 x2 x3) (4ae4a.. 4a7ef..)) x4 (proof)
Theorem 119c2.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ο . ∀ x4 . x0 = d8d01.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x3 x5 x6 = 2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Theorem fd378.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ο . ∀ x3 x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x2 x4 x5 = 2b2e3.. (f482f.. (d8d01.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) x4 x5 (proof)
Theorem 78a8c.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ο . ∀ x4 . x0 = d8d01.. x1 x2 x3 x4 ⟶ x4 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem 76505.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ο . ∀ x3 . x3 = f482f.. (d8d01.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem 2a51d.. : ∀ x0 x1 . ∀ x2 x3 : (ι → ο) → ο . ∀ x4 x5 : ι → ι → ο . ∀ x6 x7 . d8d01.. x0 x2 x4 x6 = d8d01.. x1 x3 x5 x7 ⟶ and (and (and (x0 = x1) (∀ x8 : ι → ο . (∀ x9 . x8 x9 ⟶ prim1 x9 x0) ⟶ x2 x8 = x3 x8)) (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x4 x8 x9 = x5 x8 x9)) (x6 = x7) (proof)
Theorem b3c31.. : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . ∀ x3 x4 : ι → ι → ο . ∀ x5 . (∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x0) ⟶ iff (x1 x6) (x2 x6)) ⟶ (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ iff (x3 x6 x7) (x4 x6 x7)) ⟶ d8d01.. x0 x1 x3 x5 = d8d01.. x0 x2 x4 x5 (proof)
Definition f593b.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : (ι → ο) → ο . ∀ x4 : ι → ι → ο . ∀ x5 . prim1 x5 x2 ⟶ x1 (d8d01.. x2 x3 x4 x5)) ⟶ x1 x0
Theorem 48a34.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ο . ∀ x3 . prim1 x3 x0 ⟶ f593b.. (d8d01.. x0 x1 x2 x3) (proof)
Theorem 5c3bc.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ο . ∀ x3 . f593b.. (d8d01.. x0 x1 x2 x3) ⟶ prim1 x3 x0 (proof)
Theorem b3e73.. : ∀ x0 . f593b.. x0 ⟶ x0 = d8d01.. (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Definition df6a8.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι → ο) → ι → ι . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem 69516.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι → ο) → ι → ι . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ο . ∀ x4 . (∀ x5 : (ι → ο) → ο . (∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x1) ⟶ iff (x2 x6) (x5 x6)) ⟶ ∀ x6 : ι → ι → ο . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ iff (x3 x7 x8) (x6 x7 x8)) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4) ⟶ df6a8.. (d8d01.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition 4cd9f.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι → ο) → ι → ο . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem 938bf.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι → ο) → ι → ο . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ο . ∀ x4 . (∀ x5 : (ι → ο) → ο . (∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x1) ⟶ iff (x2 x6) (x5 x6)) ⟶ ∀ x6 : ι → ι → ο . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ iff (x3 x7 x8) (x6 x7 x8)) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4) ⟶ 4cd9f.. (d8d01.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)