Proofgold Address

Param SNo^SNo : ι → ο
Param SNoLe^SNoLe : ι → ι → ο
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param sqrt_SNo_nonneg^{sqrt_SNo_nonneg} : ι → ι
Param mul_SNo^mul_SNo : ι → ι → ι
Param SNoS_^SNoS_ : ι → ι
Param SNoLev^SNoLev : ι → ι
Known SNoLev_ind^SNoLev_ind : ∀ x0 : ι → ο . (∀ x1 . SNo x1 ⟶ (∀ x2 . x2 ∈ SNoS_ (SNoLev x1) ⟶ x0 x2) ⟶ x0 x1) ⟶ ∀ x1 . SNo x1 ⟶ x0 x1
Param SNoCut^SNoCut : ι → ι → ι
Param famunion^famunion : ι → (ι → ι) → ι
Param omega^omega : ι
Param ap^ap : ι → ι → ι
Param SNo_sqrtaux^SNo_sqrtaux : ι → (ι → ι) → ι → ι
Param ordsucc^ordsucc : ι → ι
Known sqrt_SNo_nonneg_eq^{sqrt_SNo_nonneg_eq} : ∀ x0 . SNo x0 ⟶ sqrt_SNo_nonneg x0 = SNoCut (famunion omega (λ x2 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x2) 0)) (famunion omega (λ x2 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x2) 1))
Param SNoLt^SNoLt : ι → ι → ο
Param nat_p^nat_p : ι → ο
Definition SNoCutP^SNoCutP := λ x0 x1 . and (and (∀ x2 . x2 ∈ x0 ⟶ SNo x2) (∀ x2 . x2 ∈ x1 ⟶ SNo x2)) (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x1 ⟶ SNoLt x2 x3)
Known sqrt_SNo_nonneg_prop1b^{sqrt_SNo_nonneg_prop1b} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ (∀ x1 . nat_p x1 ⟶ and (∀ x2 . x2 ∈ ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 0 ⟶ and (and (SNo x2) (SNoLe 0 x2)) (SNoLt (mul_SNo x2 x2) x0)) (∀ x2 . x2 ∈ ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 1 ⟶ and (and (SNo x2) (SNoLe 0 x2)) (SNoLt x0 (mul_SNo x2 x2)))) ⟶ SNoCutP (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 0)) (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 1))
Known sqrt_SNo_nonneg_prop1a^{sqrt_SNo_nonneg_prop1a} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ (∀ x1 . x1 ∈ SNoS_ (SNoLev x0) ⟶ SNoLe 0 x1 ⟶ and (and (SNo (sqrt_SNo_nonneg x1)) (SNoLe 0 (sqrt_SNo_nonneg x1))) (mul_SNo (sqrt_SNo_nonneg x1) (sqrt_SNo_nonneg x1) = x1)) ⟶ ∀ x1 . nat_p x1 ⟶ and (∀ x2 . x2 ∈ ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 0 ⟶ and (and (SNo x2) (SNoLe 0 x2)) (SNoLt (mul_SNo x2 x2) x0)) (∀ x2 . x2 ∈ ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 1 ⟶ and (and (SNo x2) (SNoLe 0 x2)) (SNoLt x0 (mul_SNo x2 x2)))
Param binunion^binunion : ι → ι → ι
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param SNoEq_^SNoEq_ : ι → ι → ι → ο
Known SNoCutP_SNoCut_impred^{SNoCutP_SNoCut_impred} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ ∀ x2 : ο . (SNo (SNoCut x0 x1) ⟶ SNoLev (SNoCut x0 x1) ∈ ordsucc (binunion (famunion x0 (λ x3 . ordsucc (SNoLev x3))) (famunion x1 (λ x3 . ordsucc (SNoLev x3)))) ⟶ (∀ x3 . x3 ∈ x0 ⟶ SNoLt x3 (SNoCut x0 x1)) ⟶ (∀ x3 . x3 ∈ x1 ⟶ SNoLt (SNoCut x0 x1) x3) ⟶ (∀ x3 . SNo x3 ⟶ (∀ x4 . x4 ∈ x0 ⟶ SNoLt x4 x3) ⟶ (∀ x4 . x4 ∈ x1 ⟶ SNoLt x3 x4) ⟶ and (SNoLev (SNoCut x0 x1) ⊆ SNoLev x3) (SNoEq_ (SNoLev (SNoCut x0 x1)) (SNoCut x0 x1) x3)) ⟶ x2) ⟶ x2
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known sqrt_SNo_nonneg_prop1c^{sqrt_SNo_nonneg_prop1c} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ SNoCutP (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 0)) (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 1)) ⟶ (∀ x1 . x1 ∈ famunion omega (λ x2 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x2) 1) ⟶ ∀ x2 : ο . (SNo x1 ⟶ SNoLe 0 x1 ⟶ SNoLt x0 (mul_SNo x1 x1) ⟶ x2) ⟶ x2) ⟶ SNoLe 0 (SNoCut (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 0)) (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 1)))
Known SNoLt_trichotomy_or_impred^{SNoLt_trichotomy_or_impred} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 : ο . (SNoLt x0 x1 ⟶ x2) ⟶ (x0 = x1 ⟶ x2) ⟶ (SNoLt x1 x0 ⟶ x2) ⟶ x2
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Definition False^False := ∀ x0 : ο . x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known sqrt_SNo_nonneg_prop1d^{sqrt_SNo_nonneg_prop1d} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ (∀ x1 . x1 ∈ SNoS_ (SNoLev x0) ⟶ SNoLe 0 x1 ⟶ and (and (SNo (sqrt_SNo_nonneg x1)) (SNoLe 0 (sqrt_SNo_nonneg x1))) (mul_SNo (sqrt_SNo_nonneg x1) (sqrt_SNo_nonneg x1) = x1)) ⟶ SNoCutP (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 0)) (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 1)) ⟶ SNoLe 0 (SNoCut (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 0)) (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 1))) ⟶ SNoLt (mul_SNo (SNoCut (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 0)) (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 1))) (SNoCut (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 0)) (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 1)))) x0 ⟶ False
Known sqrt_SNo_nonneg_prop1e^{sqrt_SNo_nonneg_prop1e} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ (∀ x1 . x1 ∈ SNoS_ (SNoLev x0) ⟶ SNoLe 0 x1 ⟶ and (and (SNo (sqrt_SNo_nonneg x1)) (SNoLe 0 (sqrt_SNo_nonneg x1))) (mul_SNo (sqrt_SNo_nonneg x1) (sqrt_SNo_nonneg x1) = x1)) ⟶ SNoCutP (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 0)) (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 1)) ⟶ SNoLe 0 (SNoCut (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 0)) (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 1))) ⟶ SNoLt x0 (mul_SNo (SNoCut (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 0)) (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 1))) (SNoCut (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 0)) (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 1)))) ⟶ False
Known famunionE_impred^{famunionE_impred} : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ famunion x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ x0 ⟶ x2 ∈ x1 x4 ⟶ x3) ⟶ x3
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Theorem sqrt_SNo_nonneg_prop1^{sqrt_SNo_nonneg_prop1} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ and (and (SNo (sqrt_SNo_nonneg x0)) (SNoLe 0 (sqrt_SNo_nonneg x0))) (mul_SNo (sqrt_SNo_nonneg x0) (sqrt_SNo_nonneg x0) = x0) (proof)
Param ordinal^ordinal : ι → ο
Param SNo_^SNo_ : ι → ι → ο
Known SNoS_E2^SNoS_E2 : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ SNoS_ x0 ⟶ ∀ x2 : ο . (SNoLev x1 ∈ x0 ⟶ ordinal (SNoLev x1) ⟶ SNo x1 ⟶ SNo_ (SNoLev x1) x1 ⟶ x2) ⟶ x2
Known SNoLev_ordinal^{SNoLev_ordinal} : ∀ x0 . SNo x0 ⟶ ordinal (SNoLev x0)
Theorem SNo_sqrtaux_0_1_prop^{SNo_sqrtaux_0_1_prop} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ ∀ x1 . nat_p x1 ⟶ and (∀ x2 . x2 ∈ ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 0 ⟶ and (and (SNo x2) (SNoLe 0 x2)) (SNoLt (mul_SNo x2 x2) x0)) (∀ x2 . x2 ∈ ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 1 ⟶ and (and (SNo x2) (SNoLe 0 x2)) (SNoLt x0 (mul_SNo x2 x2))) (proof)
Theorem SNo_sqrtaux_0_prop^{SNo_sqrtaux_0_prop} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ ∀ x1 . nat_p x1 ⟶ ∀ x2 . x2 ∈ ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 0 ⟶ and (and (SNo x2) (SNoLe 0 x2)) (SNoLt (mul_SNo x2 x2) x0) (proof)
Theorem SNo_sqrtaux_1_prop^{SNo_sqrtaux_1_prop} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ ∀ x1 . nat_p x1 ⟶ ∀ x2 . x2 ∈ ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 1 ⟶ and (and (SNo x2) (SNoLe 0 x2)) (SNoLt x0 (mul_SNo x2 x2)) (proof)
Theorem SNo_sqrt_SNo_SNoCutP^{SNo_sqrt_SNo_SNoCutP} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ SNoCutP (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 0)) (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 1)) (proof)
Theorem SNo_sqrt_SNo_nonneg^{SNo_sqrt_SNo_nonneg} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ SNo (sqrt_SNo_nonneg x0) (proof)
Theorem sqrt_SNo_nonneg_nonneg^{sqrt_SNo_nonneg_nonneg} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ SNoLe 0 (sqrt_SNo_nonneg x0) (proof)
Theorem sqrt_SNo_nonneg_sqr^{sqrt_SNo_nonneg_sqr} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ mul_SNo (sqrt_SNo_nonneg x0) (sqrt_SNo_nonneg x0) = x0 (proof)
Known SNo_0^SNo_0 : SNo 0
Definition not^not := λ x0 : ο . x0 ⟶ False
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known Empty_eq^Empty_eq : ∀ x0 . (∀ x1 . nIn x1 x0) ⟶ x0 = 0
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Known SNoCut_0_0^SNoCut_0_0 : SNoCut 0 0 = 0
Known nat_ind^nat_ind : ∀ x0 : ι → ο . x0 0 ⟶ (∀ x1 . nat_p x1 ⟶ x0 x1 ⟶ x0 (ordsucc x1)) ⟶ ∀ x1 . nat_p x1 ⟶ x0 x1
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Param lam^Sigma : ι → (ι → ι) → ι
Param If_i^If_i : ο → ι → ι → ι
Param Sep^Sep : ι → (ι → ο) → ι
Param SNoL^SNoL : ι → ι
Definition SNoL_nonneg^SNoL_nonneg := λ x0 . Sep (SNoL x0) (SNoLe 0)
Param SNoR^SNoR : ι → ι
Known SNo_sqrtaux_0^{SNo_sqrtaux_0} : ∀ x0 . ∀ x1 : ι → ι . SNo_sqrtaux x0 x1 0 = lam 2 (λ x3 . If_i (x3 = 0) (prim5 (SNoL_nonneg x0) x1) (prim5 (SNoR x0) x1))
Known tuple_2_0_eq^tuple_2_0_eq : ∀ x0 x1 . ap (lam 2 (λ x3 . If_i (x3 = 0) x0 x1)) 0 = x0
Known SNoL_nonneg_0^{SNoL_nonneg_0} : SNoL_nonneg 0 = 0
Known Repl_Empty^Repl_Empty : ∀ x0 : ι → ι . prim5 0 x0 = 0
Known tuple_2_1_eq^tuple_2_1_eq : ∀ x0 x1 . ap (lam 2 (λ x3 . If_i (x3 = 0) x0 x1)) 1 = x1
Known SNoR_0^SNoR_0 : SNoR 0 = 0
Param SNo_sqrtauxset^{SNo_sqrtauxset} : ι → ι → ι → ι
Known SNo_sqrtaux_S^{SNo_sqrtaux_S} : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . nat_p x2 ⟶ SNo_sqrtaux x0 x1 (ordsucc x2) = lam 2 (λ x4 . If_i (x4 = 0) (binunion (ap (SNo_sqrtaux x0 x1 x2) 0) (SNo_sqrtauxset (ap (SNo_sqrtaux x0 x1 x2) 0) (ap (SNo_sqrtaux x0 x1 x2) 1) x0)) (binunion (binunion (ap (SNo_sqrtaux x0 x1 x2) 1) (SNo_sqrtauxset (ap (SNo_sqrtaux x0 x1 x2) 0) (ap (SNo_sqrtaux x0 x1 x2) 0) x0)) (SNo_sqrtauxset (ap (SNo_sqrtaux x0 x1 x2) 1) (ap (SNo_sqrtaux x0 x1 x2) 1) x0)))
Known SNo_sqrtauxset_0^{SNo_sqrtauxset_0} : ∀ x0 x1 . SNo_sqrtauxset 0 x0 x1 = 0
Known binunion_idl^binunion_idl : ∀ x0 . binunion 0 x0 = x0
Theorem sqrt_SNo_nonneg_0^{sqrt_SNo_nonneg_0} : sqrt_SNo_nonneg 0 = 0 (proof)
Known SNo_1^SNo_1 : SNo 1
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Known cases_1^cases_1 : ∀ x0 . x0 ∈ 1 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 x0
Known famunionI^famunionI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x2 ∈ x0 ⟶ x3 ∈ x1 x2 ⟶ x3 ∈ famunion x0 x1
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known nat_0^nat_0 : nat_p 0
Known In_0_1^In_0_1 : 0 ∈ 1
Known SNoL_1^SNoL_1 : SNoL 1 = 1
Known SNoR_1^SNoR_1 : SNoR 1 = 0
Known SNo_eta^SNo_eta : ∀ x0 . SNo x0 ⟶ x0 = SNoCut (SNoL x0) (SNoR x0)
Known SNoL_nonneg_1^{SNoL_nonneg_1} : SNoL_nonneg 1 = 1
Known ReplE_impred^ReplE_impred : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ prim5 x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ x0 ⟶ x2 = x1 x4 ⟶ x3) ⟶ x3
Known ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
Known SNo_sqrtauxset_0'^{SNo_sqrtauxset_0} : ∀ x0 x1 . SNo_sqrtauxset x0 0 x1 = 0
Known binunion_idr^binunion_idr : ∀ x0 . binunion x0 0 = x0
Param add_SNo^add_SNo : ι → ι → ι
Param div_SNo^div_SNo : ι → ι → ι
Known SNo_sqrtauxset_E^{SNo_sqrtauxset_E} : ∀ x0 x1 x2 x3 . x3 ∈ SNo_sqrtauxset x0 x1 x2 ⟶ ∀ x4 : ο . (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x1 ⟶ SNoLt 0 (add_SNo x5 x6) ⟶ x3 = div_SNo (add_SNo x2 (mul_SNo x5 x6)) (add_SNo x5 x6) ⟶ x4) ⟶ x4
Known add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 x0)
Theorem sqrt_SNo_nonneg_1^{sqrt_SNo_nonneg_1} : sqrt_SNo_nonneg 1 = 1 (proof)
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known SNoL_I^SNoL_I : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x1 x0 ⟶ x1 ∈ SNoL x0
Known SNoLev_0^SNoLev_0 : SNoLev 0 = 0
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known SNoLeE^SNoLeE : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ or (SNoLt x0 x1) (x0 = x1)
Known SNoLe_ref^SNoLe_ref : ∀ x0 . SNoLe x0 x0
Theorem sqrt_SNo_nonneg_0inL0^{sqrt_SNo_nonneg_0inL0} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ 0 ∈ SNoLev x0 ⟶ 0 ∈ ap (SNo_sqrtaux x0 sqrt_SNo_nonneg 0) 0 (proof)
Theorem sqrt_SNo_nonneg_Lnonempty^{sqrt_SNo_nonneg_Lnonempty} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ 0 ∈ SNoLev x0 ⟶ famunion omega (λ x2 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x2) 0) = 0 ⟶ ∀ x1 : ο . x1 (proof)
Known SNoR_I^SNoR_I : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x0 x1 ⟶ x1 ∈ SNoR x0
Known ordinal_SNoLev^{ordinal_SNoLev} : ∀ x0 . ordinal x0 ⟶ SNoLev x0 = x0
Known nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known nat_1^nat_1 : nat_p 1
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Known binunionI2^binunionI2 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ x2 ∈ binunion x0 x1
Known SNo_sqrtauxset_I^{SNo_sqrtauxset_I} : ∀ x0 x1 x2 x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x1 ⟶ SNoLt 0 (add_SNo x3 x4) ⟶ div_SNo (add_SNo x2 (mul_SNo x3 x4)) (add_SNo x3 x4) ∈ SNo_sqrtauxset x0 x1 x2
Known SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Known SNoLt_0_1^SNoLt_0_1 : SNoLt 0 1
Definition TransSet^TransSet := λ x0 . ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Known ordinal_TransSet^{ordinal_TransSet} : ∀ x0 . ordinal x0 ⟶ TransSet x0
Theorem sqrt_SNo_nonneg_Rnonempty^{sqrt_SNo_nonneg_Rnonempty} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ 1 ∈ SNoLev x0 ⟶ famunion omega (λ x2 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x2) 1) = 0 ⟶ ∀ x1 : ο . x1 (proof)