Proofgold Address

Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param real^real : ι
Param famunion^famunion : ι → (ι → ι) → ι
Param ReplSep^ReplSep : ι → (ι → ο) → (ι → ι) → ι
Param SNoLt^SNoLt : ι → ι → ο
Param add_SNo^add_SNo : ι → ι → ι
Param mul_SNo^mul_SNo : ι → ι → ι
Param recip_SNo^recip_SNo : ι → ι
Definition div_SNo^div_SNo := λ x0 x1 . mul_SNo x0 (recip_SNo x1)
Definition SNo_sqrtauxset^{SNo_sqrtauxset} := λ x0 x1 x2 . famunion x0 (λ x3 . {div_SNo (add_SNo x2 (mul_SNo x3 x4)) (add_SNo x3 x4)|x4 ∈ x1,SNoLt 0 (add_SNo x3 x4)})
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 ReplSepE_impred^{ReplSepE_impred} : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 : ι → ι . ∀ x3 . x3 ∈ ReplSep x0 x1 x2 ⟶ ∀ x4 : ο . (∀ x5 . x5 ∈ x0 ⟶ x1 x5 ⟶ x3 = x2 x5 ⟶ x4) ⟶ x4
Known real_div_SNo^real_div_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ div_SNo x0 x1 ∈ real
Known real_add_SNo^real_add_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ add_SNo x0 x1 ∈ real
Known real_mul_SNo^real_mul_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ mul_SNo x0 x1 ∈ real
Theorem SNo_sqrtauxset_real^{SNo_sqrtauxset_real} : ∀ x0 x1 x2 . x0 ⊆ real ⟶ x1 ⊆ real ⟶ x2 ∈ real ⟶ SNo_sqrtauxset x0 x1 x2 ⊆ real (proof)
Param Sep^Sep : ι → (ι → ο) → ι
Param SNoLe^SNoLe : ι → ι → ο
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known SepE1^SepE1 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x2 ∈ x0
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
Param SNo^SNo : ι → ο
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 SNo_0^SNo_0 : SNo 0
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Known real_SNo^real_SNo : ∀ x0 . x0 ∈ real ⟶ SNo x0
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Known add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
Known add_SNo_Le3^add_SNo_Le3 : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLe x0 x2 ⟶ SNoLe x1 x3 ⟶ SNoLe (add_SNo x0 x1) (add_SNo x2 x3)
Known SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Known mul_SNo_pos_pos^{mul_SNo_pos_pos} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt 0 x0 ⟶ SNoLt 0 x1 ⟶ SNoLt 0 (mul_SNo x0 x1)
Known mul_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
Known SNoLe_ref^SNoLe_ref : ∀ x0 . SNoLe x0 x0
Known mul_SNo_zeroL^{mul_SNo_zeroL} : ∀ x0 . SNo x0 ⟶ mul_SNo 0 x0 = 0
Known div_SNo_pos_pos^{div_SNo_pos_pos} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt 0 x0 ⟶ SNoLt 0 x1 ⟶ SNoLt 0 (div_SNo x0 x1)
Known div_SNo_0_num^{div_SNo_0_num} : ∀ x0 . SNo x0 ⟶ div_SNo 0 x0 = 0
Theorem SNo_sqrtauxset_real_nonneg^{SNo_sqrtauxset_real_nonneg} : ∀ x0 x1 x2 . x0 ⊆ Sep real (SNoLe 0) ⟶ x1 ⊆ Sep real (SNoLe 0) ⟶ x2 ∈ real ⟶ SNoLe 0 x2 ⟶ SNo_sqrtauxset x0 x1 x2 ⊆ Sep real (SNoLe 0) (proof)
Param SNoS_^SNoS_ : ι → ι
Param omega^omega : ι
Param sqrt_SNo_nonneg^{sqrt_SNo_nonneg} : ι → ι
Param ordinal^ordinal : ι → ο
Param SNoLev^SNoLev : ι → ι
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 omega_ordinal^{omega_ordinal} : ordinal omega
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
Known ordinal_In_Or_Subq^{ordinal_In_Or_Subq} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ or (x0 ∈ x1) (x1 ⊆ x0)
Known ordinal_Empty^{ordinal_Empty} : ordinal 0
Known SNoLev_ordinal^{SNoLev_ordinal} : ∀ x0 . SNo x0 ⟶ ordinal (SNoLev x0)
Param ordsucc^ordsucc : ι → ι
Known ordinal_1^ordinal_1 : ordinal 1
Param SNoCut^SNoCut : ι → ι → ι
Param ap^ap : ι → ι → ι
Param SNo_sqrtaux^SNo_sqrtaux : ι → (ι → ι) → 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 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 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))
Known real_SNoCut^real_SNoCut : ∀ x0 . x0 ⊆ real ⟶ ∀ x1 . x1 ⊆ real ⟶ SNoCutP x0 x1 ⟶ (x0 = 0 ⟶ ∀ x2 : ο . x2) ⟶ (x1 = 0 ⟶ ∀ x2 : ο . x2) ⟶ (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ x0) (SNoLt x2 x4) ⟶ x3) ⟶ x3) ⟶ (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ x1) (SNoLt x4 x2) ⟶ x3) ⟶ x3) ⟶ SNoCut x0 x1 ∈ real
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Known 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
Known 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
Known 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)
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_1^SNo_1 : SNo 1
Known SNo_sqrtaux_mon^{SNo_sqrtaux_mon} : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . nat_p x2 ⟶ ∀ x3 . nat_p x3 ⟶ x2 ⊆ x3 ⟶ and (ap (SNo_sqrtaux x0 x1 x2) 0 ⊆ ap (SNo_sqrtaux x0 x1 x3) 0) (ap (SNo_sqrtaux x0 x1 x2) 1 ⊆ ap (SNo_sqrtaux x0 x1 x3) 1)
Known nat_0^nat_0 : nat_p 0
Known nat_ordsucc^nat_ordsucc : ∀ x0 . nat_p x0 ⟶ nat_p (ordsucc x0)
Known Subq_Empty^Subq_Empty : ∀ x0 . 0 ⊆ x0
Param lam^Sigma : ι → (ι → ι) → ι
Param If_i^If_i : ο → ι → ι → ι
Definition SNoL^SNoL := λ x0 . {x1 ∈ SNoS_ (SNoLev x0)|SNoLt x1 x0}
Definition SNoL_nonneg^SNoL_nonneg := λ x0 . Sep (SNoL x0) (SNoLe 0)
Definition SNoR^SNoR := λ x0 . Sep (SNoS_ (SNoLev x0)) (SNoLt x0)
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_1_eq^tuple_2_1_eq : ∀ x0 x1 . ap (lam 2 (λ x3 . If_i (x3 = 0) x0 x1)) 1 = x1
Known sqrt_SNo_nonneg_1^{sqrt_SNo_nonneg_1} : sqrt_SNo_nonneg 1 = 1
Known ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
Known SNoS_I2^SNoS_I2 : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev x0 ∈ SNoLev x1 ⟶ x0 ∈ SNoS_ (SNoLev x1)
Known ordinal_SNoLev^{ordinal_SNoLev} : ∀ x0 . ordinal x0 ⟶ SNoLev x0 = x0
Definition False^False := ∀ x0 : ο . x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Param binunion^binunion : 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 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 add_SNo_1_1_2^{add_SNo_1_1_2} : add_SNo 1 1 = 2
Known SNoLt_0_2^SNoLt_0_2 : SNoLt 0 2
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 SNoLt_0_1^SNoLt_0_1 : SNoLt 0 1
Known famunionI^famunionI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x2 ∈ x0 ⟶ x3 ∈ x1 x2 ⟶ x3 ∈ famunion x0 x1
Known omega_ordsucc^{omega_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ ordsucc x0 ∈ omega
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known ordsuccI1^ordsuccI1 : ∀ x0 . x0 ⊆ ordsucc x0
Known div_mul_SNo_invL^{div_mul_SNo_invL} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ (x1 = 0 ⟶ ∀ x2 : ο . x2) ⟶ div_SNo (mul_SNo x0 x1) x1 = x0
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 x0)
Known pos_mul_SNo_Lt'^{pos_mul_SNo_Lt} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt 0 x2 ⟶ SNoLt x0 x1 ⟶ SNoLt (mul_SNo x0 x2) (mul_SNo x1 x2)
Known SNo_recip_SNo^{SNo_recip_SNo} : ∀ x0 . SNo x0 ⟶ SNo (recip_SNo x0)
Known recip_SNo_of_pos_is_pos^{recip_SNo_of_pos_is_pos} : ∀ x0 . SNo x0 ⟶ SNoLt 0 x0 ⟶ SNoLt 0 (recip_SNo x0)
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 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 SNoLtLe_tra^SNoLtLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLt x0 x2
Known SNoLeLt_tra^SNoLeLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Known add_SNo_Le1^add_SNo_Le1 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x2 ⟶ SNoLe (add_SNo x0 x1) (add_SNo x2 x1)
Known add_SNo_Lt3^add_SNo_Lt3 : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLt x0 x2 ⟶ SNoLt x1 x3 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x3)
Known 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))
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 ReplE_impred^ReplE_impred : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ prim5 x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ x0 ⟶ x2 = x1 x4 ⟶ x3) ⟶ x3
Known SNoL_E^SNoL_E : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoL x0 ⟶ ∀ x2 : ο . (SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x1 x0 ⟶ x2) ⟶ x2
Definition TransSet^TransSet := λ x0 . ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Known omega_TransSet^{omega_TransSet} : TransSet omega
Known SNoR_E^SNoR_E : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoR x0 ⟶ ∀ x2 : ο . (SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x0 x1 ⟶ x2) ⟶ x2
Known binunion_Subq_min^{binunion_Subq_min} : ∀ x0 x1 x2 . x0 ⊆ x2 ⟶ x1 ⊆ x2 ⟶ binunion x0 x1 ⊆ x2
Param iff^iff : ο → ο → ο
Definition PNoEq_^PNoEq_ := λ x0 . λ x1 x2 : ι → ο . ∀ x3 . x3 ∈ x0 ⟶ iff (x1 x3) (x2 x3)
Definition SNoEq_^SNoEq_ := λ x0 x1 x2 . PNoEq_ x0 (λ x3 . x3 ∈ x1) (λ x3 . x3 ∈ x2)
Known SNo_eq^SNo_eq : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev x0 = SNoLev x1 ⟶ SNoEq_ (SNoLev x0) x0 x1 ⟶ x0 = x1
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 iffI^iffI : ∀ x0 x1 : ο . (x0 ⟶ x1) ⟶ (x1 ⟶ x0) ⟶ iff x0 x1
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Known SNoLtI3^SNoLtI3 : ∀ x0 x1 . SNoLev x1 ∈ SNoLev x0 ⟶ SNoEq_ (SNoLev x1) x0 x1 ⟶ nIn (SNoLev x1) x0 ⟶ SNoLt x0 x1
Known SNoLev_0^SNoLev_0 : SNoLev 0 = 0
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Known real_1^real_1 : 1 ∈ real
Known SNoLev_0_eq_0^{SNoLev_0_eq_0} : ∀ x0 . SNo x0 ⟶ SNoLev x0 = 0 ⟶ x0 = 0
Known Empty_Subq_eq^{Empty_Subq_eq} : ∀ x0 . x0 ⊆ 0 ⟶ x0 = 0
Known sqrt_SNo_nonneg_0^{sqrt_SNo_nonneg_0} : sqrt_SNo_nonneg 0 = 0
Known real_0^real_0 : 0 ∈ real
Known SNoS_omega_real^{SNoS_omega_real} : SNoS_ omega ⊆ real
Known SNoS_I^SNoS_I : ∀ x0 . ordinal x0 ⟶ ∀ x1 x2 . x2 ∈ x0 ⟶ SNo_ x2 x1 ⟶ x1 ∈ SNoS_ x0
Known SNoLev_^SNoLev_ : ∀ x0 . SNo x0 ⟶ SNo_ (SNoLev x0) x0
Theorem sqrt_SNo_nonneg_SNoS_omega^{sqrt_SNo_nonneg_SNoS_omega} : ∀ x0 . x0 ∈ SNoS_ omega ⟶ SNoLe 0 x0 ⟶ sqrt_SNo_nonneg x0 ∈ real (proof)
Param minus_SNo^minus_SNo : ι → ι
Param abs_SNo^abs_SNo : ι → ι
Param eps_^eps_ : ι → ι
Known real_E^real_E : ∀ x0 . x0 ∈ real ⟶ ∀ x1 : ο . (SNo x0 ⟶ SNoLev x0 ∈ ordsucc omega ⟶ x0 ∈ SNoS_ (ordsucc omega) ⟶ SNoLt (minus_SNo omega) x0 ⟶ SNoLt x0 omega ⟶ (∀ x2 . x2 ∈ SNoS_ omega ⟶ (∀ x3 . x3 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x2 (minus_SNo x0))) (eps_ x3)) ⟶ x2 = x0) ⟶ (∀ x2 . x2 ∈ omega ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ SNoS_ omega) (and (SNoLt x4 x0) (SNoLt x0 (add_SNo x4 (eps_ x2)))) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1
Known 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)
Known ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
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 nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
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 nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known div_SNo_pos_LtR^{div_SNo_pos_LtR} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt 0 x1 ⟶ SNoLt (mul_SNo x2 x1) x0 ⟶ SNoLt x2 (div_SNo x0 x1)
Known pos_mul_SNo_Lt2^{pos_mul_SNo_Lt2} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLt 0 x0 ⟶ SNoLt 0 x1 ⟶ SNoLt x0 x2 ⟶ SNoLt x1 x3 ⟶ SNoLt (mul_SNo x0 x1) (mul_SNo x2 x3)
Known add_SNo_Lt3b^add_SNo_Lt3b : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLe x0 x2 ⟶ SNoLt x1 x3 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x3)
Known Empty_eq^Empty_eq : ∀ x0 . (∀ x1 . nIn x1 x0) ⟶ x0 = 0
Known div_SNo_pos_LtL^{div_SNo_pos_LtL} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt 0 x1 ⟶ SNoLt x0 (mul_SNo x2 x1) ⟶ SNoLt (div_SNo x0 x1) x2
Known nat_1^nat_1 : nat_p 1
Known sqrt_SNo_nonneg_nonneg^{sqrt_SNo_nonneg_nonneg} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ SNoLe 0 (sqrt_SNo_nonneg x0)
Known Subq_tra^Subq_tra : ∀ x0 x1 x2 . x0 ⊆ x1 ⟶ x1 ⊆ x2 ⟶ x0 ⊆ x2
Theorem sqrt_SNo_nonneg_real^{sqrt_SNo_nonneg_real} : ∀ x0 . x0 ∈ real ⟶ SNoLe 0 x0 ⟶ sqrt_SNo_nonneg x0 ∈ real (proof)
Param int^int : ι
Param setminus^setminus : ι → ι → ι
Param Sing^Sing : ι → ι
Definition rational^rational := {x0 ∈ real|∀ x1 : ο . (∀ x2 . and (x2 ∈ int) (∀ x3 : ο . (∀ x4 . and (x4 ∈ setminus omega (Sing 0)) (x0 = div_SNo x2 x4) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1}
Param mul_nat^mul_nat : ι → ι → ι
Known form100_1_v1^form100_1_v1 : ∀ x0 . x0 ∈ setminus omega 1 ⟶ ∀ x1 . x1 ∈ setminus omega 1 ⟶ mul_nat x0 x0 = mul_nat 2 (mul_nat x1 x1) ⟶ ∀ x2 : ο . x2
Known setminusI^setminusI : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ nIn x2 x1 ⟶ x2 ∈ setminus x0 x1
Known omega_SNoS_omega^{omega_SNoS_omega} : omega ⊆ SNoS_ omega
Known nat_2^nat_2 : nat_p 2
Known int_3_cases^int_3_cases : ∀ x0 . x0 ∈ int ⟶ ∀ x1 : ο . (∀ x2 . x2 ∈ omega ⟶ x0 = minus_SNo (ordsucc x2) ⟶ x1) ⟶ (x0 = 0 ⟶ x1) ⟶ (∀ x2 . x2 ∈ omega ⟶ x0 = ordsucc x2 ⟶ x1) ⟶ x1
Known SNo_2^SNo_2 : SNo 2
Known div_SNo_neg_pos^{div_SNo_neg_pos} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 0 ⟶ SNoLt 0 x1 ⟶ SNoLt (div_SNo x0 x1) 0
Known int_SNo^int_SNo : ∀ x0 . x0 ∈ int ⟶ SNo x0
Known minus_SNo_Lt_contra1^{minus_SNo_Lt_contra1} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt (minus_SNo x0) x1 ⟶ SNoLt (minus_SNo x1) x0
Known omega_SNo^omega_SNo : ∀ x0 . x0 ∈ omega ⟶ SNo x0
Known minus_SNo_0^minus_SNo_0 : minus_SNo 0 = 0
Known ordinal_ordsucc_SNo_eq^{ordinal_ordsucc_SNo_eq} : ∀ x0 . ordinal x0 ⟶ ordsucc x0 = add_SNo 1 x0
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 omega_nonneg^omega_nonneg : ∀ x0 . x0 ∈ omega ⟶ SNoLe 0 x0
Known neq_ordsucc_0^{neq_ordsucc_0} : ∀ x0 . ordsucc x0 = 0 ⟶ ∀ x1 : ο . x1
Known eq_1_Sing0^eq_1_Sing0 : 1 = Sing 0
Known mul_nat_mul_SNo^{mul_nat_mul_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_nat x0 x1 = mul_SNo x0 x1
Known mul_nat_p^mul_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (mul_nat x0 x1)
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 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
Known setminusE2^setminusE2 : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ nIn x2 x1
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Known mul_div_SNo_invL^{mul_div_SNo_invL} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ (x1 = 0 ⟶ ∀ x2 : ο . x2) ⟶ mul_SNo (div_SNo x0 x1) x1 = x0
Known SNo_sqrt_SNo_nonneg^{SNo_sqrt_SNo_nonneg} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ SNo (sqrt_SNo_nonneg x0)
Known setminusE1^setminusE1 : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ x2 ∈ x0
Theorem sqrt_2_irrational^{sqrt_2_irrational} : sqrt_SNo_nonneg 2 ∈ setminus real rational (proof)