Proofgold Asset

Param 0fc90.. : ι → (ι → ι) → ι
Param 4ae4a.. : ι → ι
Param 4a7ef.. : ι
Param If_i : ο → ι → ι → ι
Param eb53d.. : ι → CT2 ι
Param 1216a.. : ι → (ι → ο) → ι
Definition 59e44.. := λ x0 . λ x1 : ι → ι → ι . λ x2 : ι → ο . λ x3 x4 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) (eb53d.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (1216a.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x3 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 07fe2.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 x5 . x0 = 59e44.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem eb78a.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . ∀ x3 x4 . x0 = f482f.. (59e44.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Param e3162.. : ι → ι → ι → ι
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 35054.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ e3162.. (eb53d.. x0 x1) x2 x3 = x1 x2 x3
Theorem acf94.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 x5 . x0 = 59e44.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = e3162.. (f482f.. x0 (4ae4a.. 4a7ef..)) x6 x7 (proof)
Theorem 4cd80.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . ∀ x3 x4 x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x1 x5 x6 = e3162.. (f482f.. (59e44.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 x6 (proof)
Param decode_p : ι → ι → ο
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 931fe.. : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . prim1 x2 x0 ⟶ decode_p (1216a.. x0 x1) x2 = x1 x2
Theorem 449e8.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 x5 . x0 = 59e44.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x3 x6 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 (proof)
Theorem f200f.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . ∀ x3 x4 x5 . prim1 x5 x0 ⟶ x2 x5 = decode_p (f482f.. (59e44.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
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
Theorem 8ef7a.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 x5 . x0 = 59e44.. x1 x2 x3 x4 x5 ⟶ x4 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem a0393.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . ∀ x3 x4 . x3 = f482f.. (59e44.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
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
Theorem 540f0.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 x5 . x0 = 59e44.. x1 x2 x3 x4 x5 ⟶ x5 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem 369e2.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . ∀ x3 x4 . x4 = f482f.. (59e44.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (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 e496c.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ο . ∀ x6 x7 x8 x9 . 59e44.. x0 x2 x4 x6 x8 = 59e44.. x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 . prim1 x10 x0 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x2 x10 x11 = x3 x10 x11)) (∀ x10 . prim1 x10 x0 ⟶ x4 x10 = x5 x10)) (x6 = x7)) (x8 = x9) (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 8fdaf.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ x1 x3 x4 = x2 x3 x4) ⟶ eb53d.. x0 x1 = eb53d.. x0 x2
Theorem 0037b.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ο . ∀ x5 x6 . (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x1 x7 x8 = x2 x7 x8) ⟶ (∀ x7 . prim1 x7 x0 ⟶ iff (x3 x7) (x4 x7)) ⟶ 59e44.. x0 x1 x3 x5 x6 = 59e44.. x0 x2 x4 x5 x6 (proof)
Definition 3f728.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ι . (∀ x4 . prim1 x4 x2 ⟶ ∀ x5 . prim1 x5 x2 ⟶ prim1 (x3 x4 x5) x2) ⟶ ∀ x4 : ι → ο . ∀ x5 . prim1 x5 x2 ⟶ ∀ x6 . prim1 x6 x2 ⟶ x1 (59e44.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem 2d21c.. : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ prim1 (x1 x2 x3) x0) ⟶ ∀ x2 : ι → ο . ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ 3f728.. (59e44.. x0 x1 x2 x3 x4) (proof)
Theorem 51074.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . ∀ x3 x4 . 3f728.. (59e44.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x1 x5 x6) x0 (proof)
Theorem 19427.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . ∀ x3 x4 . 3f728.. (59e44.. x0 x1 x2 x3 x4) ⟶ prim1 x3 x0 (proof)
Theorem 223c0.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . ∀ x3 x4 . 3f728.. (59e44.. x0 x1 x2 x3 x4) ⟶ prim1 x4 x0 (proof)
Known iff_refl : ∀ x0 : ο . iff x0 x0
Theorem ca5af.. : ∀ x0 . 3f728.. x0 ⟶ x0 = 59e44.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition d34f8.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ο) → ι → ι → ι . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 02a1c.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ο) → ι → ι → ι . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 x5 . (∀ x6 : ι → ι → ι . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ x2 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ο . (∀ x8 . prim1 x8 x1 ⟶ iff (x3 x8) (x7 x8)) ⟶ x0 x1 x6 x7 x4 x5 = x0 x1 x2 x3 x4 x5) ⟶ d34f8.. (59e44.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 28d85.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ο) → ι → ι → ο . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem a9852.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ο) → ι → ι → ο . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 x5 . (∀ x6 : ι → ι → ι . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ x2 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ο . (∀ x8 . prim1 x8 x1 ⟶ iff (x3 x8) (x7 x8)) ⟶ x0 x1 x6 x7 x4 x5 = x0 x1 x2 x3 x4 x5) ⟶ 28d85.. (59e44.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Param d2155.. : ι → (ι → ι → ο) → ι
Definition dd5e6.. := λ x0 . λ x1 x2 : ι → ι . λ x3 x4 : ι → ι → ο . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) (0fc90.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (d2155.. x0 x3) (d2155.. x0 x4)))))
Theorem 25954.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = dd5e6.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 85c77.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 x5 : ι → ι → ο . x5 x0 (f482f.. (dd5e6.. x0 x1 x2 x3 x4) 4a7ef..) ⟶ x5 (f482f.. (dd5e6.. x0 x1 x2 x3 x4) 4a7ef..) x0 (proof)
Known f22ec.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 x0 ⟶ f482f.. (0fc90.. x0 x1) x2 = x1 x2
Theorem 527b1.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = dd5e6.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x6 = f482f.. (f482f.. x0 (4ae4a.. 4a7ef..)) x6 (proof)
Theorem f63f1.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x1 x5 = f482f.. (f482f.. (dd5e6.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 (proof)
Theorem ea1b8.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = dd5e6.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x3 x6 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 (proof)
Theorem e13f3.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x2 x5 = f482f.. (f482f.. (dd5e6.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Param 2b2e3.. : ι → ι → ι → ο
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 4a94a.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = dd5e6.. 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 7f895.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x3 x5 x6 = 2b2e3.. (f482f.. (dd5e6.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 x6 (proof)
Theorem ba983.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = dd5e6.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x5 x6 x7 = 2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x6 x7 (proof)
Theorem be737.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x4 x5 x6 = 2b2e3.. (f482f.. (dd5e6.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x5 x6 (proof)
Theorem 339b2.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι . ∀ x6 x7 x8 x9 : ι → ι → ο . dd5e6.. x0 x2 x4 x6 x8 = dd5e6.. x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 . prim1 x10 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 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x8 x10 x11 = x9 x10 x11) (proof)
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
Theorem 04c48.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι . ∀ x5 x6 x7 x8 : ι → ι → ο . (∀ x9 . prim1 x9 x0 ⟶ 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 ⟶ ∀ x10 . prim1 x10 x0 ⟶ iff (x7 x9 x10) (x8 x9 x10)) ⟶ dd5e6.. x0 x1 x3 x5 x7 = dd5e6.. x0 x2 x4 x6 x8 (proof)
Definition 04214.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι . (∀ x4 . prim1 x4 x2 ⟶ prim1 (x3 x4) x2) ⟶ ∀ x4 : ι → ι . (∀ x5 . prim1 x5 x2 ⟶ prim1 (x4 x5) x2) ⟶ ∀ x5 x6 : ι → ι → ο . x1 (dd5e6.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem 2c935.. : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . prim1 x2 x0 ⟶ prim1 (x1 x2) x0) ⟶ ∀ x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) x0) ⟶ ∀ x3 x4 : ι → ι → ο . 04214.. (dd5e6.. x0 x1 x2 x3 x4) (proof)
Theorem 0e89e.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . 04214.. (dd5e6.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x1 x5) x0 (proof)
Theorem 6e401.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . 04214.. (dd5e6.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x2 x5) x0 (proof)
Theorem d6ad5.. : ∀ x0 . 04214.. x0 ⟶ x0 = dd5e6.. (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))) (proof)
Definition 834d5.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → (ι → ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem e167e.. : ∀ x0 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → (ι → ι → ο) → ι . ∀ x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . (∀ x6 : ι → ι . (∀ x7 . prim1 x7 x1 ⟶ 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 → ι → ο . (∀ x10 . prim1 x10 x1 ⟶ ∀ x11 . prim1 x11 x1 ⟶ iff (x5 x10 x11) (x9 x10 x11)) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5) ⟶ 834d5.. (dd5e6.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 3233e.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → (ι → ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem 86968.. : ∀ x0 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → (ι → ι → ο) → ο . ∀ x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . (∀ x6 : ι → ι . (∀ x7 . prim1 x7 x1 ⟶ 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 → ι → ο . (∀ x10 . prim1 x10 x1 ⟶ ∀ x11 . prim1 x11 x1 ⟶ iff (x5 x10 x11) (x9 x10 x11)) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5) ⟶ 3233e.. (dd5e6.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 50ab0.. := λ x0 . λ x1 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..) (0fc90.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (d2155.. x0 x3) (1216a.. x0 x4)))))
Theorem ef2c1.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . x0 = 50ab0.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 56472.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . x0 = f482f.. (50ab0.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Theorem 72955.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . x0 = 50ab0.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x6 = f482f.. (f482f.. x0 (4ae4a.. 4a7ef..)) x6 (proof)
Theorem ebcb1.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x1 x5 = f482f.. (f482f.. (50ab0.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 (proof)
Theorem 67bb9.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . x0 = 50ab0.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x3 x6 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 (proof)
Theorem 71dc2.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x2 x5 = f482f.. (f482f.. (50ab0.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Theorem 8eb97.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . x0 = 50ab0.. 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 f21b1.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x3 x5 x6 = 2b2e3.. (f482f.. (50ab0.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 x6 (proof)
Theorem 1ad09.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . x0 = 50ab0.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x5 x6 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x6 (proof)
Theorem 41cdc.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x4 x5 = decode_p (f482f.. (50ab0.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x5 (proof)
Theorem 39c4e.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι . ∀ x6 x7 : ι → ι → ο . ∀ x8 x9 : ι → ο . 50ab0.. x0 x2 x4 x6 x8 = 50ab0.. x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 . prim1 x10 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)
Theorem 7e3c3.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι . ∀ x5 x6 : ι → ι → ο . ∀ x7 x8 : ι → ο . (∀ x9 . prim1 x9 x0 ⟶ 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)) ⟶ 50ab0.. x0 x1 x3 x5 x7 = 50ab0.. x0 x2 x4 x6 x8 (proof)
Definition 3db62.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι . (∀ x4 . prim1 x4 x2 ⟶ prim1 (x3 x4) x2) ⟶ ∀ x4 : ι → ι . (∀ x5 . prim1 x5 x2 ⟶ prim1 (x4 x5) x2) ⟶ ∀ x5 : ι → ι → ο . ∀ x6 : ι → ο . x1 (50ab0.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem de947.. : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . prim1 x2 x0 ⟶ prim1 (x1 x2) x0) ⟶ ∀ x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) x0) ⟶ ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . 3db62.. (50ab0.. x0 x1 x2 x3 x4) (proof)
Theorem c61d8.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . 3db62.. (50ab0.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x1 x5) x0 (proof)
Theorem d45f1.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . 3db62.. (50ab0.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x2 x5) x0 (proof)
Theorem d963b.. : ∀ x0 . 3db62.. x0 ⟶ x0 = 50ab0.. (f482f.. x0 4a7ef..) (f482f.. (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 f52b0.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → (ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (f482f.. (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 035a3.. : ∀ x0 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → (ι → ο) → ι . ∀ x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . (∀ x6 : ι → ι . (∀ x7 . prim1 x7 x1 ⟶ 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) ⟶ f52b0.. (50ab0.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 8fa66.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → (ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (f482f.. (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 bc231.. : ∀ x0 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → (ι → ο) → ο . ∀ x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . (∀ x6 : ι → ι . (∀ x7 . prim1 x7 x1 ⟶ 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) ⟶ 8fa66.. (50ab0.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 2cece.. := λ x0 . λ x1 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..) (0fc90.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (d2155.. x0 x3) x4))))
Theorem c81aa.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . x0 = 2cece.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 18d49.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . x0 = f482f.. (2cece.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Theorem fd872.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . x0 = 2cece.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x6 = f482f.. (f482f.. x0 (4ae4a.. 4a7ef..)) x6 (proof)
Theorem d9dd8.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . prim1 x5 x0 ⟶ x1 x5 = f482f.. (f482f.. (2cece.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 (proof)
Theorem e6fc5.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . x0 = 2cece.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x3 x6 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 (proof)
Theorem b9f3a.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . prim1 x5 x0 ⟶ x2 x5 = f482f.. (f482f.. (2cece.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Theorem 2c5e3.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . x0 = 2cece.. 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 4fe0c.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x3 x5 x6 = 2b2e3.. (f482f.. (2cece.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 x6 (proof)
Theorem 89916.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . x0 = 2cece.. x1 x2 x3 x4 x5 ⟶ x5 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem 14a76.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . x4 = f482f.. (2cece.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem abd90.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι . ∀ x6 x7 : ι → ι → ο . ∀ x8 x9 . 2cece.. x0 x2 x4 x6 x8 = 2cece.. x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 . prim1 x10 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 379e6.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι . ∀ x5 x6 : ι → ι → ο . ∀ x7 . (∀ x8 . prim1 x8 x0 ⟶ 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)) ⟶ 2cece.. x0 x1 x3 x5 x7 = 2cece.. x0 x2 x4 x6 x7 (proof)
Definition a8a42.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι . (∀ x4 . prim1 x4 x2 ⟶ prim1 (x3 x4) x2) ⟶ ∀ x4 : ι → ι . (∀ x5 . prim1 x5 x2 ⟶ prim1 (x4 x5) x2) ⟶ ∀ x5 : ι → ι → ο . ∀ x6 . prim1 x6 x2 ⟶ x1 (2cece.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem db973.. : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . prim1 x2 x0 ⟶ prim1 (x1 x2) x0) ⟶ ∀ x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) x0) ⟶ ∀ x3 : ι → ι → ο . ∀ x4 . prim1 x4 x0 ⟶ a8a42.. (2cece.. x0 x1 x2 x3 x4) (proof)
Theorem 66a05.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . a8a42.. (2cece.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x1 x5) x0 (proof)
Theorem b42a7.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . a8a42.. (2cece.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x2 x5) x0 (proof)
Theorem a259e.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . a8a42.. (2cece.. x0 x1 x2 x3 x4) ⟶ prim1 x4 x0 (proof)
Theorem 58b86.. : ∀ x0 . a8a42.. x0 ⟶ x0 = 2cece.. (f482f.. x0 4a7ef..) (f482f.. (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 8740e.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → ι → ι . x1 (f482f.. x0 4a7ef..) (f482f.. (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 931ca.. : ∀ x0 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → ι → ι . ∀ x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . (∀ x6 : ι → ι . (∀ x7 . prim1 x7 x1 ⟶ 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) ⟶ 8740e.. (2cece.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 61d02.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → ι → ο . x1 (f482f.. x0 4a7ef..) (f482f.. (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 ee478.. : ∀ x0 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → ι → ο . ∀ x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . (∀ x6 : ι → ι . (∀ x7 . prim1 x7 x1 ⟶ 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) ⟶ 61d02.. (2cece.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)