Proofgold Address

Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param omega^omega : ι
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Param add_nat^add_nat : ι → ι → ι
Param mul_nat^mul_nat : ι → ι → ι
Definition b3e62..^{equiv_nat_mod} := λ x0 x1 x2 . and (and (and (x0 ∈ omega) (x1 ∈ omega)) (x2 ∈ omega)) (or (∀ x3 : ο . (∀ x4 . and (x4 ∈ omega) (add_nat x0 (mul_nat x4 x2) = x1) ⟶ x3) ⟶ x3) (∀ x3 : ο . (∀ x4 . and (x4 ∈ omega) (add_nat x1 (mul_nat x4 x2) = x0) ⟶ x3) ⟶ x3))
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Param ordsucc^ordsucc : ι → ι
Definition divides_nat^divides_nat := λ x0 x1 . and (and (x0 ∈ omega) (x1 ∈ omega)) (∀ x2 : ο . (∀ x3 . and (x3 ∈ omega) (mul_nat x0 x3 = x1) ⟶ x2) ⟶ x2)
Definition prime_nat^prime_nat := λ x0 . and (and (x0 ∈ omega) (1 ∈ x0)) (∀ x1 . x1 ∈ omega ⟶ divides_nat x1 x0 ⟶ or (x1 = 1) (x1 = x0))
Known c14f2.. : not (prime_nat 4)
Param nat_p^nat_p : ι → ο
Known 92637.. : ∀ x0 . nat_p x0 ⟶ mul_nat x0 x0 = x0 ⟶ or (x0 = 0) (x0 = 1)
Param equip^equip : ι → ι → ο
Param exp_nat^exp_nat : ι → ι → ι
Param Pi^Pi : ι → (ι → ι) → ι
Definition setexp^setexp := λ x0 x1 . Pi x1 (λ x2 . x0)
Known 17a87.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ equip (exp_nat x0 x1) (setexp x0 x1)
Param SNo^SNo : ι → ο
Param mul_SNo^mul_SNo : ι → ι → ι
Known 24b4c.. : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ mul_SNo x0 (mul_SNo x1 (mul_SNo x2 x3)) = mul_SNo x1 (mul_SNo x2 (mul_SNo x0 x3))
Param add_SNo^add_SNo : ι → ι → ι
Param minus_SNo^minus_SNo : ι → ι
Param lam^Sigma : ι → (ι → ι) → ι
Param If_i^If_i : ο → ι → ι → ι
Param ap^ap : ι → ι → ι
Param SNoLt^SNoLt : ι → ι → ο
Known 91770.. : ∀ x0 x1 x2 : ι → ι → ι → ι . ∀ x3 x4 x5 : ι → ι → ι → ο . (∀ x6 x7 x8 . x3 x6 x7 x8 ⟶ x0 x6 x7 x8 = add_SNo x6 (mul_SNo 2 x8)) ⟶ (∀ x6 x7 x8 . x3 x6 x7 x8 ⟶ x1 x6 x7 x8 = x8) ⟶ (∀ x6 x7 x8 . x3 x6 x7 x8 ⟶ x2 x6 x7 x8 = add_SNo x7 (minus_SNo (add_SNo x6 x8))) ⟶ (∀ x6 x7 x8 . x4 x6 x7 x8 ⟶ x0 x6 x7 x8 = add_SNo (mul_SNo 2 x7) (minus_SNo x6)) ⟶ (∀ x6 x7 x8 . x4 x6 x7 x8 ⟶ x1 x6 x7 x8 = x7) ⟶ (∀ x6 x7 x8 . x4 x6 x7 x8 ⟶ x2 x6 x7 x8 = add_SNo x6 (add_SNo x8 (minus_SNo x7))) ⟶ (∀ x6 x7 x8 . x5 x6 x7 x8 ⟶ x0 x6 x7 x8 = add_SNo x6 (minus_SNo (mul_SNo 2 x7))) ⟶ (∀ x6 x7 x8 . x5 x6 x7 x8 ⟶ x1 x6 x7 x8 = add_SNo x6 (add_SNo x8 (minus_SNo x7))) ⟶ (∀ x6 x7 x8 . x5 x6 x7 x8 ⟶ x2 x6 x7 x8 = x7) ⟶ (∀ x6 x7 x8 . add_SNo x6 x8 ∈ x7 ⟶ x3 x6 x7 x8) ⟶ (∀ x6 x7 x8 . x3 x6 x7 x8 ⟶ add_SNo x6 x8 ∈ x7) ⟶ (∀ x6 x7 x8 . x7 ∈ add_SNo x6 x8 ⟶ x6 ∈ mul_SNo 2 x7 ⟶ x4 x6 x7 x8) ⟶ (∀ x6 x7 x8 . x4 x6 x7 x8 ⟶ x7 ∈ add_SNo x6 x8) ⟶ (∀ x6 x7 x8 . x4 x6 x7 x8 ⟶ x6 ∈ mul_SNo 2 x7) ⟶ (∀ x6 x7 x8 . x7 ∈ add_SNo x6 x8 ⟶ mul_SNo 2 x7 ∈ x6 ⟶ x5 x6 x7 x8) ⟶ (∀ x6 x7 x8 . x5 x6 x7 x8 ⟶ x7 ∈ add_SNo x6 x8) ⟶ (∀ x6 x7 x8 . x5 x6 x7 x8 ⟶ mul_SNo 2 x7 ∈ x6) ⟶ ∀ x6 . x6 ∈ omega ⟶ ∀ x7 . equip x7 x6 ⟶ (∀ x8 . x8 ∈ x7 ⟶ lam 3 (λ x10 . If_i (x10 = 0) (ap x8 0) (If_i (x10 = 1) (ap x8 1) (ap x8 2))) = x8) ⟶ (∀ x8 x9 x10 . lam 3 (λ x11 . If_i (x11 = 0) x8 (If_i (x11 = 1) x9 x10)) ∈ x7 ⟶ ∀ x11 : ο . (x8 ∈ omega ⟶ x9 ∈ omega ⟶ x10 ∈ omega ⟶ SNoLt 0 x8 ⟶ SNoLt 0 x9 ⟶ SNoLt 0 x10 ⟶ (add_SNo x8 x10 = x9 ⟶ ∀ x12 : ο . x12) ⟶ (x8 = mul_SNo 2 x9 ⟶ ∀ x12 : ο . x12) ⟶ lam 3 (λ x12 . If_i (x12 = 0) x8 (If_i (x12 = 1) x10 x9)) ∈ x7 ⟶ x11) ⟶ x11) ⟶ (∀ x8 x9 x10 x11 . x11 ∈ omega ⟶ ∀ x12 . x12 ∈ omega ⟶ ∀ x13 . x13 ∈ omega ⟶ lam 3 (λ x14 . If_i (x14 = 0) x8 (If_i (x14 = 1) x9 x10)) ∈ x7 ⟶ add_SNo (mul_SNo x11 x11) (mul_SNo 4 (mul_SNo x12 x13)) = add_SNo (mul_SNo x8 x8) (mul_SNo 4 (mul_SNo x9 x10)) ⟶ lam 3 (λ x14 . If_i (x14 = 0) x11 (If_i (x14 = 1) x12 x13)) ∈ x7) ⟶ ∀ x8 x9 x10 . lam 3 (λ x11 . If_i (x11 = 0) x8 (If_i (x11 = 1) x9 x10)) ∈ x7 ⟶ x0 x8 x9 x10 = x8 ⟶ x1 x8 x9 x10 = x9 ⟶ x2 x8 x9 x10 = x10 ⟶ (∀ x11 x12 x13 . lam 3 (λ x14 . If_i (x14 = 0) x11 (If_i (x14 = 1) x12 x13)) ∈ x7 ⟶ x0 x11 x12 x13 = x11 ⟶ x1 x11 x12 x13 = x12 ⟶ x2 x11 x12 x13 = x13 ⟶ and (and (x11 = x8) (x12 = x9)) (x13 = x10)) ⟶ ∀ x11 : ο . (∀ x12 . (∀ x13 : ο . (∀ x14 . lam 3 (λ x15 . If_i (x15 = 0) x12 (If_i (x15 = 1) x14 x14)) ∈ x7 ⟶ x13) ⟶ x13) ⟶ x11) ⟶ x11
Known c188a.. : SNoLt 1 4
Param ordinal^ordinal : ι → ο
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param Sep^Sep : ι → (ι → ο) → ι
Definition finite^finite := λ x0 . ∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (equip x0 x2) ⟶ x1) ⟶ x1
Known Subq_finite^Subq_finite : ∀ x0 . finite x0 ⟶ ∀ x1 . x1 ⊆ x0 ⟶ finite x1
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known 4f402..^exp_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (exp_nat x0 x1)
Known nat_3^nat_3 : nat_p 3
Known equip_sym^equip_sym : ∀ x0 x1 . equip x0 x1 ⟶ equip x1 x0
Param SNoLe^SNoLe : ι → ι → ο
Known Pi_eta^Pi_eta : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ Pi x0 x1 ⟶ lam x0 (ap x2) = x2
Known lam_Pi^lam_Pi : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1 x3) ⟶ lam x0 x2 ∈ Pi x0 x1
Known cases_3^cases_3 : ∀ x0 . x0 ∈ 3 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 1 ⟶ x1 2 ⟶ x1 x0
Known ordinal_In_Or_Subq^{ordinal_In_Or_Subq} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ or (x0 ∈ x1) (x1 ⊆ x0)
Known nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 x0)
Known SNoLeLt_tra^SNoLeLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known SNoLe_tra^SNoLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLe x0 x2
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Known mul_SNo_oneR^mul_SNo_oneR : ∀ x0 . SNo x0 ⟶ mul_SNo x0 1 = x0
Known nonneg_mul_SNo_Le^{nonneg_mul_SNo_Le} : ∀ x0 x1 x2 . SNo x0 ⟶ SNoLe 0 x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x1 x2 ⟶ SNoLe (mul_SNo x0 x1) (mul_SNo x0 x2)
Known ordinal_Subq_SNoLe^{ordinal_Subq_SNoLe} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ x0 ⊆ x1 ⟶ SNoLe x0 x1
Known ordinal_Empty^{ordinal_Empty} : ordinal 0
Known Subq_Empty^Subq_Empty : ∀ x0 . 0 ⊆ x0
Known SNo_1^SNo_1 : SNo 1
Known mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
Known add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
Known add_SNo_Lt2^add_SNo_Lt2 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x1 x2 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x0 x2)
Known SNo_0^SNo_0 : SNo 0
Known SNoLt_tra^SNoLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known SNoLt_0_1^SNoLt_0_1 : SNoLt 0 1
Known SNoLtLe_tra^SNoLtLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLt x0 x2
Known 48da5.. : SNo 4
Known nat_p_SNo^nat_p_SNo : ∀ x0 . nat_p x0 ⟶ SNo x0
Known nat_4^nat_4 : nat_p 4
Known pos_mul_SNo_Lt^{pos_mul_SNo_Lt} : ∀ x0 x1 x2 . SNo x0 ⟶ SNoLt 0 x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x1 x2 ⟶ SNoLt (mul_SNo x0 x1) (mul_SNo x0 x2)
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Known add_SNo_Lt1^add_SNo_Lt1 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x2 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x1)
Known omega_SNo^omega_SNo : ∀ x0 . x0 ∈ omega ⟶ SNo x0
Known nat_1^nat_1 : nat_p 1
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known mul_nat_mul_SNo^{mul_nat_mul_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_nat x0 x1 = mul_SNo x0 x1
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Known mul_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
Known cases_1^cases_1 : ∀ x0 . x0 ∈ 1 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 x0
Known mul_SNo_zeroL^{mul_SNo_zeroL} : ∀ x0 . SNo x0 ⟶ mul_SNo 0 x0 = 0
Known neq_4_1^neq_4_1 : 4 = 1 ⟶ ∀ x0 : ο . x0
Known SepE2^SepE2 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x1 x2
Known mul_SNo_In_omega^{mul_SNo_In_omega} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_SNo x0 x1 ∈ omega
Known In_2_3^In_2_3 : 2 ∈ 3
Known In_1_3^In_1_3 : 1 ∈ 3
Known In_0_3^In_0_3 : 0 ∈ 3
Known If_i_0^If_i_0 : ∀ x0 : ο . ∀ x1 x2 . not x0 ⟶ If_i x0 x1 x2 = x2
Known In_no2cycle^In_no2cycle : ∀ x0 x1 . x0 ∈ x1 ⟶ x1 ∈ x0 ⟶ False
Known If_i_1^If_i_1 : ∀ x0 : ο . ∀ x1 x2 . x0 ⟶ If_i x0 x1 x2 = x1
Known add_nat_add_SNo^{add_nat_add_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ add_nat x0 x1 = add_SNo x0 x1
Known add_SNo_Le2^add_SNo_Le2 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x1 x2 ⟶ SNoLe (add_SNo x0 x1) (add_SNo x0 x2)
Known ordinal_In_SNoLt^{ordinal_In_SNoLt} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ SNoLt x1 x0
Known Pi_ext^Pi_ext : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ Pi x0 x1 ⟶ ∀ x3 . x3 ∈ Pi x0 x1 ⟶ (∀ x4 . x4 ∈ x0 ⟶ ap x2 x4 = ap x3 x4) ⟶ x2 = x3
Known tuple_3_in_A_3^{tuple_3_in_A_3} : ∀ x0 x1 x2 x3 . x0 ∈ x3 ⟶ x1 ∈ x3 ⟶ x2 ∈ x3 ⟶ lam 3 (λ x4 . If_i (x4 = 0) x0 (If_i (x4 = 1) x1 x2)) ∈ setexp x3 3
Known tuple_3_0_eq^tuple_3_0_eq : ∀ x0 x1 x2 . ap (lam 3 (λ x4 . If_i (x4 = 0) x0 (If_i (x4 = 1) x1 x2))) 0 = x0
Known tuple_3_1_eq^tuple_3_1_eq : ∀ x0 x1 x2 . ap (lam 3 (λ x4 . If_i (x4 = 0) x0 (If_i (x4 = 1) x1 x2))) 1 = x1
Known tuple_3_2_eq^tuple_3_2_eq : ∀ x0 x1 x2 . ap (lam 3 (λ x4 . If_i (x4 = 0) x0 (If_i (x4 = 1) x1 x2))) 2 = x2
Known ordinal_trichotomy_or_impred^{ordinal_trichotomy_or_impred} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ ∀ x2 : ο . (x0 ∈ x1 ⟶ x2) ⟶ (x0 = x1 ⟶ x2) ⟶ (x1 ∈ x0 ⟶ x2) ⟶ x2
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Known SNo_2^SNo_2 : SNo 2
Known SNoLt_1_2^SNoLt_1_2 : SNoLt 1 2
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known mul_SNo_oneL^mul_SNo_oneL : ∀ x0 . SNo x0 ⟶ mul_SNo 1 x0 = x0
Known add_SNo_1_1_2^{add_SNo_1_1_2} : add_SNo 1 1 = 2
Known add_SNo_minus_R2^{add_SNo_minus_R2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo (add_SNo x0 x1) (minus_SNo x1) = x0
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
Known add_SNo_minus_SNo_prop2^{add_SNo_minus_SNo_prop2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 (add_SNo (minus_SNo x0) x1) = x1
Known c8949.. : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = 0 ⟶ or (x0 = 0) (x1 = 0)
Known add_SNo_cancel_L^{add_SNo_cancel_L} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 x1 = add_SNo x0 x2 ⟶ x1 = x2
Known neq_2_0^neq_2_0 : 2 = 0 ⟶ ∀ x0 : ο . x0
Known mul_SNo_nonzero_cancel^{mul_SNo_nonzero_cancel_L} : ∀ x0 x1 x2 . SNo x0 ⟶ (x0 = 0 ⟶ ∀ x3 : ο . x3) ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x0 x1 = mul_SNo x0 x2 ⟶ x1 = x2
Known neq_4_0^neq_4_0 : 4 = 0 ⟶ ∀ x0 : ο . x0
Known add_SNo_In_omega^{add_SNo_In_omega} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ add_SNo x0 x1 ∈ omega
Known mul_SNo_distrL^{mul_SNo_distrL} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x0 (add_SNo x1 x2) = add_SNo (mul_SNo x0 x1) (mul_SNo x0 x2)
Known mul_SNo_com_3_0_1^{mul_SNo_com_3_0_1} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x0 (mul_SNo x1 x2) = mul_SNo x1 (mul_SNo x0 x2)
Known d67ed.. : ∀ x0 . SNo x0 ⟶ mul_SNo 2 x0 = add_SNo x0 x0
Known add_SNo_minus_R2'^{add_SNo_minus_R2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo (add_SNo x0 (minus_SNo x1)) x1 = x0
Known nat_2^nat_2 : nat_p 2
Known mul_SNo_com_4_inner_mid^{mul_SNo_com_4_inner_mid} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ mul_SNo (mul_SNo x0 x1) (mul_SNo x2 x3) = mul_SNo (mul_SNo x0 x2) (mul_SNo x1 x3)
Known ecc46.. : mul_SNo 2 2 = 4
Known ordinal_SNoLt_In^{ordinal_SNoLt_In} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ SNoLt x0 x1 ⟶ x0 ∈ x1
Known nat_inv^nat_inv : ∀ x0 . nat_p x0 ⟶ or (x0 = 0) (∀ x1 : ο . (∀ x2 . and (nat_p x2) (x0 = ordsucc x2) ⟶ x1) ⟶ x1)
Known ordinal_ordsucc^{ordinal_ordsucc} : ∀ x0 . ordinal x0 ⟶ ordinal (ordsucc x0)
Known nat_0_in_ordsucc^{nat_0_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ 0 ∈ ordsucc x0
Known In_1_2^In_1_2 : 1 ∈ 2
Known SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
Known 8b4bf.. : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo (add_SNo x0 x1) (add_SNo x0 x1) = add_SNo (mul_SNo x0 x0) (add_SNo (mul_SNo 2 (mul_SNo x0 x1)) (mul_SNo x1 x1))
Known mul_SNo_distrR^{mul_SNo_distrR} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo (add_SNo x0 x1) x2 = add_SNo (mul_SNo x0 x2) (mul_SNo x1 x2)
Known mul_SNo_assoc^{mul_SNo_assoc} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x0 (mul_SNo x1 x2) = mul_SNo (mul_SNo x0 x1) x2
Known ap_Pi^ap_Pi : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x2 ∈ Pi x0 x1 ⟶ x3 ∈ x0 ⟶ ap x2 x3 ∈ x1 x3
Known Sep_Subq^Sep_Subq : ∀ x0 . ∀ x1 : ι → ο . Sep x0 x1 ⊆ x0
Known mul_nat_1R^mul_nat_1R : ∀ x0 . mul_nat x0 1 = x0
Theorem a1a52.. : ∀ x0 . x0 ∈ omega ⟶ prime_nat x0 ⟶ b3e62.. x0 1 4 ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (∀ x3 : ο . (∀ x4 . and (x4 ∈ omega) (x0 = add_SNo (mul_SNo x2 x2) (mul_SNo x4 x4)) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1 (proof)
Theorem 09af7.. : ∀ x0 . x0 ∈ omega ⟶ prime_nat x0 ⟶ b3e62.. x0 1 4 ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (∀ x3 : ο . (∀ x4 . and (x4 ∈ omega) (x0 = add_nat (mul_nat x2 x2) (mul_nat x4 x4)) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1 (proof)