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 d2155.. : ι → (ι → ι → ο) → ι
Definition 4d5a4.. := λ x0 . λ x1 : ι → ι → ι . 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..) (eb53d.. x0 x1) (If_i (x4 = 4ae4a.. (4ae4a.. 4a7ef..)) (d2155.. x0 x2) 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 55222.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . x0 = 4d5a4.. x1 x2 x3 x4 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem ca5f3.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 . x0 = f482f.. (4d5a4.. x0 x1 x2 x3) 4a7ef.. (proof)
Param e3162.. : ι → ι → ι → ι
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 35054.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ e3162.. (eb53d.. x0 x1) x2 x3 = x1 x2 x3
Theorem 58d81.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . x0 = 4d5a4.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x5 x6 = e3162.. (f482f.. x0 (4ae4a.. 4a7ef..)) x5 x6 (proof)
Theorem 1a809.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x1 x4 x5 = e3162.. (f482f.. (4d5a4.. x0 x1 x2 x3) (4ae4a.. 4a7ef..)) x4 x5 (proof)
Param 2b2e3.. : ι → ι → ι → ο
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 67416.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ 2b2e3.. (d2155.. x0 x1) x2 x3 = x1 x2 x3
Theorem 0ad7c.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . x0 = 4d5a4.. 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 9bf50.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x2 x4 x5 = 2b2e3.. (f482f.. (4d5a4.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) x4 x5 (proof)
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
Theorem 06e3e.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . x0 = 4d5a4.. x1 x2 x3 x4 ⟶ x4 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem da287.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 . x3 = f482f.. (4d5a4.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (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 7d9db.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ι → ο . ∀ x6 x7 . 4d5a4.. x0 x2 x4 x6 = 4d5a4.. x1 x3 x5 x7 ⟶ and (and (and (x0 = x1) (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x2 x8 x9 = x3 x8 x9)) (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x4 x8 x9 = x5 x8 x9)) (x6 = x7) (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 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 7e5c6.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ι → ο . ∀ x5 . (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x1 x6 x7 = x2 x6 x7) ⟶ (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ iff (x3 x6 x7) (x4 x6 x7)) ⟶ 4d5a4.. x0 x1 x3 x5 = 4d5a4.. x0 x2 x4 x5 (proof)
Definition 95b66.. := 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 → ι → ο . ∀ x5 . prim1 x5 x2 ⟶ x1 (4d5a4.. x2 x3 x4 x5)) ⟶ x1 x0
Theorem 91375.. : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ prim1 (x1 x2 x3) x0) ⟶ ∀ x2 : ι → ι → ο . ∀ x3 . prim1 x3 x0 ⟶ 95b66.. (4d5a4.. x0 x1 x2 x3) (proof)
Theorem 90ad0.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 . 95b66.. (4d5a4.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x1 x4 x5) x0 (proof)
Theorem b2090.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 . 95b66.. (4d5a4.. x0 x1 x2 x3) ⟶ prim1 x3 x0 (proof)
Known iff_refl : ∀ x0 : ο . iff x0 x0
Theorem 0dd31.. : ∀ x0 . 95b66.. x0 ⟶ x0 = 4d5a4.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Definition b1285.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ο) → ι → ι . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem 5de34.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ο) → ι → ι . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . (∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = x5 x6 x7) ⟶ ∀ 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) ⟶ b1285.. (4d5a4.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition 83f8d.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ο) → ι → ο . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem acc69.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ο) → ι → ο . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . (∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = x5 x6 x7) ⟶ ∀ 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) ⟶ 83f8d.. (4d5a4.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Param 1216a.. : ι → (ι → ο) → ι
Definition 60b2c.. := λ x0 . λ x1 : ι → ι → ι . λ x2 : ι → ο . λ x3 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x4 . If_i (x4 = 4a7ef..) x0 (If_i (x4 = 4ae4a.. 4a7ef..) (eb53d.. x0 x1) (If_i (x4 = 4ae4a.. (4ae4a.. 4a7ef..)) (1216a.. x0 x2) x3)))
Theorem adcff.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 . x0 = 60b2c.. x1 x2 x3 x4 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 1d6f8.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . ∀ x3 . x0 = f482f.. (60b2c.. x0 x1 x2 x3) 4a7ef.. (proof)
Theorem 68ea5.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 . x0 = 60b2c.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x5 x6 = e3162.. (f482f.. x0 (4ae4a.. 4a7ef..)) x5 x6 (proof)
Theorem ee439.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . ∀ x3 x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x1 x4 x5 = e3162.. (f482f.. (60b2c.. x0 x1 x2 x3) (4ae4a.. 4a7ef..)) x4 x5 (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 208da.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 . x0 = 60b2c.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x3 x5 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Theorem 6382b.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . ∀ x3 x4 . prim1 x4 x0 ⟶ x2 x4 = decode_p (f482f.. (60b2c.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) x4 (proof)
Theorem 1e34f.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 . x0 = 60b2c.. x1 x2 x3 x4 ⟶ x4 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem 3dd89.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . ∀ x3 . x3 = f482f.. (60b2c.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem ccbda.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ο . ∀ x6 x7 . 60b2c.. x0 x2 x4 x6 = 60b2c.. x1 x3 x5 x7 ⟶ and (and (and (x0 = x1) (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x2 x8 x9 = x3 x8 x9)) (∀ x8 . prim1 x8 x0 ⟶ x4 x8 = x5 x8)) (x6 = x7) (proof)
Known ee7ef.. : ∀ x0 . ∀ x1 x2 : ι → ο . (∀ x3 . prim1 x3 x0 ⟶ iff (x1 x3) (x2 x3)) ⟶ 1216a.. x0 x1 = 1216a.. x0 x2
Theorem 14318.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ο . ∀ x5 . (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x1 x6 x7 = x2 x6 x7) ⟶ (∀ x6 . prim1 x6 x0 ⟶ iff (x3 x6) (x4 x6)) ⟶ 60b2c.. x0 x1 x3 x5 = 60b2c.. x0 x2 x4 x5 (proof)
Definition 1c3d6.. := 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 ⟶ x1 (60b2c.. x2 x3 x4 x5)) ⟶ x1 x0
Theorem 06a3b.. : ∀ 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 ⟶ 1c3d6.. (60b2c.. x0 x1 x2 x3) (proof)
Theorem 5928f.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . ∀ x3 . 1c3d6.. (60b2c.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x1 x4 x5) x0 (proof)
Theorem b32dd.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . ∀ x3 . 1c3d6.. (60b2c.. x0 x1 x2 x3) ⟶ prim1 x3 x0 (proof)
Theorem cc1ac.. : ∀ x0 . 1c3d6.. x0 ⟶ x0 = 60b2c.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Definition 6f238.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ο) → ι → ι . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem fe575.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ο) → ι → ι . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 . (∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = x5 x6 x7) ⟶ ∀ x6 : ι → ο . (∀ x7 . prim1 x7 x1 ⟶ iff (x3 x7) (x6 x7)) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4) ⟶ 6f238.. (60b2c.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition dd052.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ο) → ι → ο . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem 9d5e6.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ο) → ι → ο . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 . (∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = x5 x6 x7) ⟶ ∀ x6 : ι → ο . (∀ x7 . prim1 x7 x1 ⟶ iff (x3 x7) (x6 x7)) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4) ⟶ dd052.. (60b2c.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition 942b6.. := λ x0 . λ x1 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..) (0fc90.. x0 x1) (If_i (x4 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. x0 x2) (d2155.. x0 x3))))
Theorem 23bd6.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . x0 = 942b6.. x1 x2 x3 x4 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem d3666.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . x4 x0 (f482f.. (942b6.. x0 x1 x2 x3) 4a7ef..) ⟶ x4 (f482f.. (942b6.. x0 x1 x2 x3) 4a7ef..) x0 (proof)
Known f22ec.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 x0 ⟶ f482f.. (0fc90.. x0 x1) x2 = x1 x2
Theorem 85096.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . x0 = 942b6.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x2 x5 = f482f.. (f482f.. x0 (4ae4a.. 4a7ef..)) x5 (proof)
Theorem e5917.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . prim1 x4 x0 ⟶ x1 x4 = f482f.. (f482f.. (942b6.. x0 x1 x2 x3) (4ae4a.. 4a7ef..)) x4 (proof)
Theorem 0f3fa.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . x0 = 942b6.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x3 x5 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Theorem c00a9.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . prim1 x4 x0 ⟶ x2 x4 = f482f.. (f482f.. (942b6.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) x4 (proof)
Theorem 428c8.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . x0 = 942b6.. 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 ffd34.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x3 x4 x5 = 2b2e3.. (f482f.. (942b6.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x4 x5 (proof)
Theorem a0ae9.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι . ∀ x6 x7 : ι → ι → ο . 942b6.. x0 x2 x4 x6 = 942b6.. x1 x3 x5 x7 ⟶ and (and (and (x0 = x1) (∀ x8 . prim1 x8 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)
Known 4402a.. : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ x1 x3 = x2 x3) ⟶ 0fc90.. x0 x1 = 0fc90.. x0 x2
Theorem 83984.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι . ∀ x5 x6 : ι → ι → ο . (∀ x7 . prim1 x7 x0 ⟶ 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)) ⟶ 942b6.. x0 x1 x3 x5 = 942b6.. x0 x2 x4 x6 (proof)
Definition 24591.. := 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 . x1 (942b6.. x2 x3 x4 x5)) ⟶ x1 x0
Theorem d6b4d.. : ∀ 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 . 24591.. (942b6.. x0 x1 x2 x3) (proof)
Theorem 4abe4.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . 24591.. (942b6.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x1 x4) x0 (proof)
Theorem 4957e.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . 24591.. (942b6.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x4) x0 (proof)
Theorem a43d0.. : ∀ x0 . 24591.. x0 ⟶ x0 = 942b6.. (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition 5ad38.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → (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..)))))
Theorem 637a6.. : ∀ x0 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → ι . ∀ x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . (∀ x5 : ι → ι . (∀ x6 . prim1 x6 x1 ⟶ 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) ⟶ 5ad38.. (942b6.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition 8f2fa.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → (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..)))))
Theorem 2fd40.. : ∀ x0 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → ο . ∀ x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . (∀ x5 : ι → ι . (∀ x6 . prim1 x6 x1 ⟶ 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) ⟶ 8f2fa.. (942b6.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition 7ba51.. := λ x0 . λ x1 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..) (0fc90.. x0 x1) (If_i (x4 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. x0 x2) (1216a.. x0 x3))))
Theorem 8f50d.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ο . x0 = 7ba51.. x1 x2 x3 x4 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem dfafe.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ο . x0 = f482f.. (7ba51.. x0 x1 x2 x3) 4a7ef.. (proof)
Theorem 92643.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ο . x0 = 7ba51.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x2 x5 = f482f.. (f482f.. x0 (4ae4a.. 4a7ef..)) x5 (proof)
Theorem 6f593.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ο . ∀ x4 . prim1 x4 x0 ⟶ x1 x4 = f482f.. (f482f.. (7ba51.. x0 x1 x2 x3) (4ae4a.. 4a7ef..)) x4 (proof)
Theorem 6425a.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ο . x0 = 7ba51.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x3 x5 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Theorem 6c23d.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ο . ∀ x4 . prim1 x4 x0 ⟶ x2 x4 = f482f.. (f482f.. (7ba51.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) x4 (proof)
Theorem 1a4cb.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ο . x0 = 7ba51.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x4 x5 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 (proof)
Theorem f0b6a.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ο . ∀ x4 . prim1 x4 x0 ⟶ x3 x4 = decode_p (f482f.. (7ba51.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x4 (proof)
Theorem bb207.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι . ∀ x6 x7 : ι → ο . 7ba51.. x0 x2 x4 x6 = 7ba51.. x1 x3 x5 x7 ⟶ and (and (and (x0 = x1) (∀ x8 . prim1 x8 x0 ⟶ x2 x8 = x3 x8)) (∀ x8 . prim1 x8 x0 ⟶ x4 x8 = x5 x8)) (∀ x8 . prim1 x8 x0 ⟶ x6 x8 = x7 x8) (proof)
Theorem 986ac.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι . ∀ x5 x6 : ι → ο . (∀ x7 . prim1 x7 x0 ⟶ x1 x7 = x2 x7) ⟶ (∀ x7 . prim1 x7 x0 ⟶ x3 x7 = x4 x7) ⟶ (∀ x7 . prim1 x7 x0 ⟶ iff (x5 x7) (x6 x7)) ⟶ 7ba51.. x0 x1 x3 x5 = 7ba51.. x0 x2 x4 x6 (proof)
Definition bfe00.. := 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 → ο . x1 (7ba51.. x2 x3 x4 x5)) ⟶ x1 x0
Theorem 5c0c3.. : ∀ 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 → ο . bfe00.. (7ba51.. x0 x1 x2 x3) (proof)
Theorem 9b099.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ο . bfe00.. (7ba51.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x1 x4) x0 (proof)
Theorem 630ad.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ο . bfe00.. (7ba51.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x4) x0 (proof)
Theorem 78290.. : ∀ x0 . bfe00.. x0 ⟶ x0 = 7ba51.. (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition 84815.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → (ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 83897.. : ∀ x0 : ι → (ι → ι) → (ι → ι) → (ι → ο) → ι . ∀ x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ο . (∀ x5 : ι → ι . (∀ x6 . prim1 x6 x1 ⟶ 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) ⟶ 84815.. (7ba51.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition b41b9.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → (ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 114cc.. : ∀ x0 : ι → (ι → ι) → (ι → ι) → (ι → ο) → ο . ∀ x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ο . (∀ x5 : ι → ι . (∀ x6 . prim1 x6 x1 ⟶ 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) ⟶ b41b9.. (7ba51.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition 5f184.. := λ x0 . λ x1 x2 : ι → ι . λ x3 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x4 . If_i (x4 = 4a7ef..) x0 (If_i (x4 = 4ae4a.. 4a7ef..) (0fc90.. x0 x1) (If_i (x4 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. x0 x2) x3)))
Theorem 590ce.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 . x0 = 5f184.. x1 x2 x3 x4 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem f7a25.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 . x0 = f482f.. (5f184.. x0 x1 x2 x3) 4a7ef.. (proof)
Theorem 00164.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 . x0 = 5f184.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x2 x5 = f482f.. (f482f.. x0 (4ae4a.. 4a7ef..)) x5 (proof)
Theorem 5254a.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 . prim1 x4 x0 ⟶ x1 x4 = f482f.. (f482f.. (5f184.. x0 x1 x2 x3) (4ae4a.. 4a7ef..)) x4 (proof)
Theorem ffb92.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 . x0 = 5f184.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x3 x5 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Theorem e662d.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 . prim1 x4 x0 ⟶ x2 x4 = f482f.. (f482f.. (5f184.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) x4 (proof)
Theorem b9667.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 . x0 = 5f184.. x1 x2 x3 x4 ⟶ x4 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem b9d33.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 . x3 = f482f.. (5f184.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem 6f78f.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι . ∀ x6 x7 . 5f184.. x0 x2 x4 x6 = 5f184.. x1 x3 x5 x7 ⟶ and (and (and (x0 = x1) (∀ x8 . prim1 x8 x0 ⟶ x2 x8 = x3 x8)) (∀ x8 . prim1 x8 x0 ⟶ x4 x8 = x5 x8)) (x6 = x7) (proof)
Theorem 87289.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι . ∀ x5 . (∀ x6 . prim1 x6 x0 ⟶ x1 x6 = x2 x6) ⟶ (∀ x6 . prim1 x6 x0 ⟶ x3 x6 = x4 x6) ⟶ 5f184.. x0 x1 x3 x5 = 5f184.. x0 x2 x4 x5 (proof)
Definition 883b9.. := 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 . prim1 x5 x2 ⟶ x1 (5f184.. x2 x3 x4 x5)) ⟶ x1 x0
Theorem 6f842.. : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . prim1 x2 x0 ⟶ prim1 (x1 x2) x0) ⟶ ∀ x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) x0) ⟶ ∀ x3 . prim1 x3 x0 ⟶ 883b9.. (5f184.. x0 x1 x2 x3) (proof)
Theorem 38e53.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 . 883b9.. (5f184.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x1 x4) x0 (proof)
Theorem d9111.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 . 883b9.. (5f184.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x4) x0 (proof)
Theorem 4e03a.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 . 883b9.. (5f184.. x0 x1 x2 x3) ⟶ prim1 x3 x0 (proof)
Theorem fb3e6.. : ∀ x0 . 883b9.. x0 ⟶ x0 = 5f184.. (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Definition 7c58a.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → ι → ι . x1 (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem 74618.. : ∀ x0 : ι → (ι → ι) → (ι → ι) → ι → ι . ∀ x1 . ∀ x2 x3 : ι → ι . ∀ x4 . (∀ x5 : ι → ι . (∀ x6 . prim1 x6 x1 ⟶ 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) ⟶ 7c58a.. (5f184.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition 98b20.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → ι → ο . x1 (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem 36c69.. : ∀ x0 : ι → (ι → ι) → (ι → ι) → ι → ο . ∀ x1 . ∀ x2 x3 : ι → ι . ∀ x4 . (∀ x5 : ι → ι . (∀ x6 . prim1 x6 x1 ⟶ 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) ⟶ 98b20.. (5f184.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)