Proofgold Signed Transaction

Param SNo^SNo : ι → ο
Param SNoLev^SNoLev : ι → ι
Param SNoR^SNoR : ι → ι
Param ordinal^ordinal : ι → ο
Known ordinal_SNoR^ordinal_SNoR : ∀ x0 . ordinal x0 ⟶ SNoR x0 = 0
Known SNoLev_ordinal^{SNoLev_ordinal} : ∀ x0 . SNo x0 ⟶ ordinal (SNoLev x0)
Theorem dd834..^{Conj_ordinal_SNoR__1__0} : ∀ x0 . SNo x0 ⟶ SNoLev x0 = x0 ⟶ SNoR x0 = 0 (proof)
Param ordsucc^ordsucc : ι → ι
Known ordsuccI2^ordsuccI2 : ∀ x0 . x0 ∈ ordsucc x0
Theorem a558a..^{Conj_SNoL_1__1__0} : ∀ x0 . SNoLev x0 ∈ 1 ⟶ 0 = x0 ⟶ x0 ∈ 1 (proof)
Param SNoCutP^SNoCutP : ι → ι → ο
Param SNoL^SNoL : ι → ι
Param SNoCut^SNoCut : ι → ι → ι
Known SNo_eta^SNo_eta : ∀ x0 . SNo x0 ⟶ x0 = SNoCut (SNoL x0) (SNoR x0)
Theorem 6cf6d..^{Conj_SNo_eta__5__1} : ∀ x0 . SNo x0 ⟶ SNoCutP (SNoL x0) (SNoR x0) ⟶ x0 = SNoCut (SNoL x0) (SNoR x0) (proof)
Param omega^omega : ι
Param nat_p^nat_p : ι → ο
Param mul_SNo^mul_SNo : ι → ι → ι
Param minus_SNo^minus_SNo : ι → ι
Param int^int : ι
Known int_mul_SNo^int_mul_SNo : ∀ x0 . x0 ∈ int ⟶ ∀ x1 . x1 ∈ int ⟶ mul_SNo x0 x1 ∈ int
Known int_minus_SNo^{int_minus_SNo} : ∀ x0 . x0 ∈ int ⟶ minus_SNo x0 ∈ int
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known Subq_omega_int^{Subq_omega_int} : omega ⊆ int
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Theorem 164f1..^{Conj_int_mul_SNo__3__2} : ∀ x0 x1 . x0 ∈ omega ⟶ SNo x0 ⟶ nat_p x1 ⟶ mul_SNo (minus_SNo x0) (minus_SNo x1) ∈ int (proof)
Param add_SNo^add_SNo : ι → ι → ι
Param eps_^eps_ : ι → ι
Known eps_ordsucc_half_add^{eps_ordsucc_half_add} : ∀ x0 . nat_p x0 ⟶ add_SNo (eps_ (ordsucc x0)) (eps_ (ordsucc x0)) = eps_ x0
Theorem ca45c..^{Conj_eps_ordsucc_half_add__11__1} : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega ⟶ add_SNo (eps_ (ordsucc x0)) (eps_ (ordsucc x0)) = eps_ x0 (proof)
Param CSNo^CSNo : ι → ο
Param add_CSNo^add_CSNo : ι → ι → ι
Known 80a5b..^add_CSNo_com : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ add_CSNo x0 x1 = add_CSNo x1 x0
Theorem a44cb..^{Conj_add_CSNo_com__1__2} : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo (add_CSNo x1 x0) ⟶ add_CSNo x0 x1 = add_CSNo x1 x0 (proof)
Param SNoLt^SNoLt : ι → ι → ο
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 x0)
Theorem d43ff..^{Conj_SNo_pos_eps_Le__1__3} : ∀ x0 x1 . SNoLt 0 x1 ⟶ ordinal (SNoLev x1) ⟶ SNo x1 ⟶ x1 = 0 ⟶ ∀ x2 : ο . x2 (proof)
Theorem d43ff..^{Conj_SNo_pos_eps_Le__1__3} : ∀ x0 x1 . SNoLt 0 x1 ⟶ ordinal (SNoLev x1) ⟶ SNo x1 ⟶ x1 = 0 ⟶ ∀ x2 : ο . x2 (proof)
Known minus_add_SNo_distr^{minus_add_SNo_distr} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ minus_SNo (add_SNo x0 x1) = add_SNo (minus_SNo x0) (minus_SNo x1)
Theorem 75d78..^{Conj_minus_add_SNo_distr__3__2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (minus_SNo x1) ⟶ minus_SNo (add_SNo x0 x1) = add_SNo (minus_SNo x0) (minus_SNo x1) (proof)
Known add_SNo_ordinal_ordinal^{add_SNo_ordinal_ordinal} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . ordinal x1 ⟶ ordinal (add_SNo x0 x1)
Theorem d2f6b..^{Conj_add_SNo_ordinal_ordinal__4__2} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ SNo x1 ⟶ ordinal (add_SNo x0 x1) (proof)
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param SNoElts_^SNoElts_ : ι → ι
Param exactly1of2^exactly1of2 : ο → ο → ο
Param binunion^binunion : ι → ι → ι
Param Sing^Sing : ι → ι
Definition SetAdjoin^SetAdjoin := λ x0 x1 . binunion x0 (Sing x1)
Definition SNo_^SNo_ := λ x0 x1 . and (x1 ⊆ SNoElts_ x0) (∀ x2 . x2 ∈ x0 ⟶ exactly1of2 (SetAdjoin x2 (Sing 1) ∈ x1) (x2 ∈ x1))
Param binintersect^binintersect : ι → ι → ι
Known SNo_SNo^SNo_SNo : ∀ x0 . ordinal x0 ⟶ ∀ x1 . SNo_ x0 x1 ⟶ SNo x1
Known ordinal_Hered^{ordinal_Hered} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordinal x1
Theorem 61455..^{Conj_restr_SNo__1__2} : ∀ x0 x1 . SNo x0 ⟶ x1 ∈ SNoLev x0 ⟶ SNo_ x1 (binintersect x0 (SNoElts_ x1)) ⟶ SNo (binintersect x0 (SNoElts_ x1)) (proof)
Param add_nat^add_nat : ι → ι → ι
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Theorem b53aa..^{Conj_add_nat_add_SNo__1__1} : ∀ x0 . ordinal x0 ⟶ (∀ x1 . nat_p x1 ⟶ add_nat x0 x1 = add_SNo x0 x1) ⟶ ∀ x1 . x1 ∈ omega ⟶ add_nat x0 x1 = add_SNo x0 x1 (proof)
Theorem df8dd..^{Conj_add_SNo_ordinal_ordinal__3__3} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ SNo x0 ⟶ SNo (add_SNo x0 x1) ⟶ ordinal (add_SNo x0 x1) (proof)
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)
Known b9e15..^{not_ordinal_Sing2} : not (ordinal (Sing 2))
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known Sing2_notin_SingSing1^{Sing2_notin_SingSing1} : nIn (Sing 2) (Sing (Sing 1))
Theorem 34c67..^{Conj_ctagged_eqE_Subq__1__1} : ∀ x0 x1 . Sing 2 ∈ x0 ⟶ ordinal x1 ⟶ not (Sing 2 ∈ SetAdjoin x1 (Sing 1)) (proof)
Param SNoS_^SNoS_ : ι → ι
Theorem 70a69..^{Conj_add_SNo_com__2__3} : ∀ x0 x1 x2 . SNo x0 ⟶ (∀ x3 . x3 ∈ SNoS_ (SNoLev x0) ⟶ add_SNo x3 x1 = add_SNo x1 x3) ⟶ SNo x2 ⟶ x2 ∈ SNoS_ (SNoLev x0) ⟶ add_SNo x2 x1 = add_SNo x1 x2 (proof)
Theorem 70a69..^{Conj_add_SNo_com__2__3} : ∀ x0 x1 x2 . SNo x0 ⟶ (∀ x3 . x3 ∈ SNoS_ (SNoLev x0) ⟶ add_SNo x3 x1 = add_SNo x1 x3) ⟶ SNo x2 ⟶ x2 ∈ SNoS_ (SNoLev x0) ⟶ add_SNo x2 x1 = add_SNo x1 x2 (proof)
Known SNo__eps_^SNo__eps_ : ∀ x0 . x0 ∈ omega ⟶ SNo_ (ordsucc x0) (eps_ x0)
Theorem dcf98..^{Conj_SNo__eps___3__3} : ∀ x0 x1 x2 . nat_p x0 ⟶ x1 ∈ ordsucc x0 ⟶ nat_p x2 ⟶ nat_p x1 ⟶ exactly1of2 (SetAdjoin x1 (Sing 1) ∈ eps_ x0) (x1 ∈ eps_ x0) (proof)
Param UPair^UPair : ι → ι → ι
Param If_i^If_i : ο → ι → ι → ι
Known UPairE^UPairE : ∀ x0 x1 x2 . x0 ∈ UPair x1 x2 ⟶ or (x0 = x1) (x0 = x2)
Known If_i_1^If_i_1 : ∀ x0 : ο . ∀ x1 x2 . x0 ⟶ If_i x0 x1 x2 = x1
Known Self_In_Power^{Self_In_Power} : ∀ x0 . x0 ∈ prim4 x0
Known ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
Known If_i_0^If_i_0 : ∀ x0 : ο . ∀ x1 x2 . not x0 ⟶ If_i x0 x1 x2 = x2
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Known Empty_In_Power^{Empty_In_Power} : ∀ x0 . 0 ∈ prim4 x0
Theorem d2732..^{Conj_ZF_UPair_closed__1__1} : ∀ x0 x1 x2 . x2 ∈ UPair x0 x1 ⟶ If_i (x0 ∈ 0) x0 x1 ∈ {If_i (x0 ∈ x3) x0 x1|x3 ∈ prim4 (prim4 x0)} ⟶ x2 ∈ {If_i (x0 ∈ x3) x0 x1|x3 ∈ prim4 (prim4 x0)} (proof)
Known b63e9..^{add_CSNo_assoc} : ∀ x0 x1 x2 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo x2 ⟶ add_CSNo x0 (add_CSNo x1 x2) = add_CSNo (add_CSNo x0 x1) x2
Theorem 31641..^{Conj_add_CSNo_assoc__2__4} : ∀ x0 x1 x2 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo x2 ⟶ CSNo (add_CSNo x1 x2) ⟶ CSNo (add_CSNo x0 (add_CSNo x1 x2)) ⟶ add_CSNo x0 (add_CSNo x1 x2) = add_CSNo (add_CSNo x0 x1) x2 (proof)
Param real^real : ι
Known pos_real_recip_ex^{pos_real_recip_ex} : ∀ x0 . x0 ∈ real ⟶ SNoLt 0 x0 ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ real) (mul_SNo x0 x2 = 1) ⟶ x1) ⟶ x1
Theorem f4ed2..^{Conj_pos_real_recip_ex__2__4} : ∀ x0 x1 . x0 ∈ real ⟶ SNoLt 0 x0 ⟶ SNo x0 ⟶ x1 ∈ omega ⟶ eps_ x1 ∈ real ⟶ SNoLt 0 (mul_SNo (eps_ x1) x0) ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ real) (mul_SNo x0 x3 = 1) ⟶ x2) ⟶ x2 (proof)
Param SNoLe^SNoLe : ι → ι → ο
Theorem 3aec6..^{Conj_SNo_approx_real_rep__1__1} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 x1 ⟶ x1 ∈ SNoS_ omega ⟶ SNoLt 0 (add_SNo x1 (minus_SNo x0)) ⟶ not (∀ x2 : ο . (∀ x3 . and (x3 ∈ omega) (SNoLe (add_SNo x0 (eps_ x3)) x1) ⟶ x2) ⟶ x2) ⟶ x1 = x0 ⟶ ∀ x2 : ο . x2 (proof)
Theorem 5ece1..^{Conj_add_SNo_prop1__4__2} : ∀ x0 x1 x2 . SNo x1 ⟶ (∀ x3 . x3 ∈ SNoS_ (SNoLev x1) ⟶ and (and (and (and (and (SNo (add_SNo x0 x3)) (∀ x4 . x4 ∈ SNoL x0 ⟶ SNoLt (add_SNo x4 x3) (add_SNo x0 x3))) (∀ x4 . x4 ∈ SNoR x0 ⟶ SNoLt (add_SNo x0 x3) (add_SNo x4 x3))) (∀ x4 . x4 ∈ SNoL x3 ⟶ SNoLt (add_SNo x0 x4) (add_SNo x0 x3))) (∀ x4 . x4 ∈ SNoR x3 ⟶ SNoLt (add_SNo x0 x3) (add_SNo x0 x4))) (SNoCutP (binunion {add_SNo x4 x3|x4 ∈ SNoL x0} (prim5 (SNoL x3) (add_SNo x0))) (binunion {add_SNo x4 x3|x4 ∈ SNoR x0} (prim5 (SNoR x3) (add_SNo x0))))) ⟶ SNoLev x2 ∈ SNoLev x1 ⟶ x2 ∈ SNoS_ (SNoLev x1) ⟶ and (and (and (and (and (SNo (add_SNo x0 x2)) (∀ x3 . x3 ∈ SNoL x0 ⟶ SNoLt (add_SNo x3 x2) (add_SNo x0 x2))) (∀ x3 . x3 ∈ SNoR x0 ⟶ SNoLt (add_SNo x0 x2) (add_SNo x3 x2))) (∀ x3 . x3 ∈ SNoL x2 ⟶ SNoLt (add_SNo x0 x3) (add_SNo x0 x2))) (∀ x3 . x3 ∈ SNoR x2 ⟶ SNoLt (add_SNo x0 x2) (add_SNo x0 x3))) (SNoCutP (binunion {add_SNo x3 x2|x3 ∈ SNoL x0} (prim5 (SNoL x2) (add_SNo x0))) (binunion {add_SNo x3 x2|x3 ∈ SNoR x0} (prim5 (SNoR x2) (add_SNo x0)))) (proof)
Theorem 5ece1..^{Conj_add_SNo_prop1__4__2} : ∀ x0 x1 x2 . SNo x1 ⟶ (∀ x3 . x3 ∈ SNoS_ (SNoLev x1) ⟶ and (and (and (and (and (SNo (add_SNo x0 x3)) (∀ x4 . x4 ∈ SNoL x0 ⟶ SNoLt (add_SNo x4 x3) (add_SNo x0 x3))) (∀ x4 . x4 ∈ SNoR x0 ⟶ SNoLt (add_SNo x0 x3) (add_SNo x4 x3))) (∀ x4 . x4 ∈ SNoL x3 ⟶ SNoLt (add_SNo x0 x4) (add_SNo x0 x3))) (∀ x4 . x4 ∈ SNoR x3 ⟶ SNoLt (add_SNo x0 x3) (add_SNo x0 x4))) (SNoCutP (binunion {add_SNo x4 x3|x4 ∈ SNoL x0} (prim5 (SNoL x3) (add_SNo x0))) (binunion {add_SNo x4 x3|x4 ∈ SNoR x0} (prim5 (SNoR x3) (add_SNo x0))))) ⟶ SNoLev x2 ∈ SNoLev x1 ⟶ x2 ∈ SNoS_ (SNoLev x1) ⟶ and (and (and (and (and (SNo (add_SNo x0 x2)) (∀ x3 . x3 ∈ SNoL x0 ⟶ SNoLt (add_SNo x3 x2) (add_SNo x0 x2))) (∀ x3 . x3 ∈ SNoR x0 ⟶ SNoLt (add_SNo x0 x2) (add_SNo x3 x2))) (∀ x3 . x3 ∈ SNoL x2 ⟶ SNoLt (add_SNo x0 x3) (add_SNo x0 x2))) (∀ x3 . x3 ∈ SNoR x2 ⟶ SNoLt (add_SNo x0 x2) (add_SNo x0 x3))) (SNoCutP (binunion {add_SNo x3 x2|x3 ∈ SNoL x0} (prim5 (SNoL x2) (add_SNo x0))) (binunion {add_SNo x3 x2|x3 ∈ SNoR x0} (prim5 (SNoR x2) (add_SNo x0)))) (proof)
Param TransSet^TransSet : ι → ο
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known ordinal_SNoLev_max_2^{ordinal_SNoLev_max_2} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ ordsucc x0 ⟶ SNoLe x1 x0
Theorem 63f7a..^{Conj_ordinal_SNoLev_max_2__5__0} : ∀ x0 x1 . TransSet x0 ⟶ SNo x1 ⟶ SNo x0 ⟶ SNoLev x0 = x0 ⟶ SNoLev x1 = x0 ⟶ not (SNoLe x1 x0) ⟶ not (∀ x2 . ordinal x2 ⟶ x2 ∈ x0 ⟶ x2 ∈ x1) (proof)
Param PNoLt^PNoLt : ι → (ι → ο) → ι → (ι → ο) → ο
Param PNoEq_^PNoEq_ : ι → (ι → ο) → (ι → ο) → ο
Known PNoLt_trichotomy_or^{PNoLt_trichotomy_or} : ∀ x0 x1 . ∀ x2 x3 : ι → ο . ordinal x0 ⟶ ordinal x1 ⟶ or (or (PNoLt x0 x2 x1 x3) (and (x0 = x1) (PNoEq_ x0 x2 x3))) (PNoLt x1 x3 x0 x2)
Theorem 07010..^{Conj_PNoLt_trichotomy_or__7__2} : ∀ x0 x1 . ∀ x2 x3 : ι → ο . ordinal x0 ⟶ ordinal x1 ⟶ TransSet x1 ⟶ ordinal (binintersect x0 x1) ⟶ or (or (PNoLt x0 x2 x1 x3) (and (x0 = x1) (PNoEq_ x0 x2 x3))) (PNoLt x1 x3 x0 x2) (proof)
Known minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0
Theorem a064b..^{Conj_minus_SNo_invol__8__2} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ (∀ x2 . x2 ∈ x0 ⟶ minus_SNo (minus_SNo x2) = x2) ⟶ (∀ x2 . x2 ∈ x0 ⟶ SNo x2) ⟶ (∀ x2 . x2 ∈ x1 ⟶ SNo x2) ⟶ SNo (SNoCut x0 x1) ⟶ minus_SNo (minus_SNo (SNoCut x0 x1)) = SNoCut x0 x1 (proof)
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Known add_SNo_minus_Lt1b^{add_SNo_minus_Lt1b} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 (add_SNo x2 x1) ⟶ SNoLt (add_SNo x0 (minus_SNo x1)) x2
Known SNo_0^SNo_0 : SNo 0
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Theorem 6cfb5..^{Conj_add_SNo_minus_SNo_linv__9__5} : ∀ x0 x1 x2 . SNo x0 ⟶ (∀ x3 . x3 ∈ SNoS_ (SNoLev x0) ⟶ add_SNo (minus_SNo x3) x3 = 0) ⟶ SNo (minus_SNo x0) ⟶ x1 = add_SNo (minus_SNo x0) x2 ⟶ SNo x2 ⟶ SNoLt x2 x0 ⟶ SNo (minus_SNo x2) ⟶ SNoLt x1 0 (proof)
Theorem 687fa..^{Conj_minus_SNo_invol__8__0} : ∀ x0 x1 . (∀ x2 . x2 ∈ x0 ⟶ minus_SNo (minus_SNo x2) = x2) ⟶ (∀ x2 . x2 ∈ x1 ⟶ minus_SNo (minus_SNo x2) = x2) ⟶ (∀ x2 . x2 ∈ x0 ⟶ SNo x2) ⟶ (∀ x2 . x2 ∈ x1 ⟶ SNo x2) ⟶ SNo (SNoCut x0 x1) ⟶ minus_SNo (minus_SNo (SNoCut x0 x1)) = SNoCut x0 x1 (proof)
Known eps_ordsucc_Lt : ∀ x0 . x0 ∈ omega ⟶ SNoLt (eps_ (ordsucc x0)) (eps_ x0)
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known SNoLeLt_tra^SNoLeLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
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 SNo_eps_^SNo_eps_ : ∀ x0 . x0 ∈ omega ⟶ SNo (eps_ x0)
Known omega_ordsucc^{omega_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ ordsucc x0 ∈ omega
Theorem 47fc9..^{Conj_eps_ordsucc_half_add__7__0} : ∀ x0 x1 . x0 ∈ omega ⟶ SNo (eps_ (ordsucc x0)) ⟶ SNo x1 ⟶ SNoLev x1 ∈ ordsucc (ordsucc x0) ⟶ SNoLt x1 (eps_ (ordsucc x0)) ⟶ SNoLe x1 0 ⟶ and (SNoLt (add_SNo x1 (eps_ (ordsucc x0))) (eps_ x0)) (SNoLt (add_SNo (eps_ (ordsucc x0)) x1) (eps_ x0)) (proof)
Known PNoLt_irref^PNoLt_irref : ∀ x0 . ∀ x1 : ι → ο . not (PNoLt x0 x1 x0 x1)
Known PNoLt_tra^PNoLt_tra : ∀ x0 x1 x2 . ordinal x0 ⟶ ordinal x1 ⟶ ordinal x2 ⟶ ∀ x3 x4 x5 : ι → ο . PNoLt x0 x3 x1 x4 ⟶ PNoLt x1 x4 x2 x5 ⟶ PNoLt x0 x3 x2 x5
Theorem e1d83..^{Conj_PNo_rel_imv_ex__7__4} : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 . ordinal x0 ⟶ PNoEq_ x0 x1 (λ x4 . or (x1 x4) (x4 = x0)) ⟶ x3 ∈ x0 ⟶ PNoEq_ x3 (λ x4 . or (x1 x4) (x4 = x0)) x2 ⟶ ordinal x3 ⟶ PNoLt x3 x2 x0 x1 ⟶ not (PNoLt x0 x1 x3 x2) (proof)
Theorem ec2d3..^{Conj_PNo_rel_imv_ex__16__2} : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 . ordinal x0 ⟶ PNoEq_ x0 (λ x4 . and (x1 x4) (x4 = x0 ⟶ ∀ x5 : ο . x5)) x1 ⟶ PNoEq_ x3 x2 (λ x4 . and (x1 x4) (x4 = x0 ⟶ ∀ x5 : ο . x5)) ⟶ and (x1 x3) (x3 = x0 ⟶ ∀ x4 : ο . x4) ⟶ ordinal x3 ⟶ PNoLt x0 x1 x3 x2 ⟶ not (PNoLt x3 x2 x0 x1) (proof)
Theorem 712f0..^{Conj_mul_SNo_eq__19__1} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . (∀ x6 . x6 ∈ SNoS_ (SNoLev x0) ⟶ ∀ x7 . SNo x7 ⟶ x2 x6 x7 = x3 x6 x7) ⟶ SNo x5 ⟶ x2 x4 x1 = x3 x4 x1 ⟶ x2 x0 x5 = x3 x0 x5 ⟶ x2 x4 x5 = x3 x4 x5 ⟶ add_SNo (x2 x4 x1) (add_SNo (x2 x0 x5) (minus_SNo (x2 x4 x5))) = add_SNo (x3 x4 x1) (add_SNo (x3 x0 x5) (minus_SNo (x3 x4 x5))) (proof)
Theorem 712f0..^{Conj_mul_SNo_eq__19__1} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . (∀ x6 . x6 ∈ SNoS_ (SNoLev x0) ⟶ ∀ x7 . SNo x7 ⟶ x2 x6 x7 = x3 x6 x7) ⟶ SNo x5 ⟶ x2 x4 x1 = x3 x4 x1 ⟶ x2 x0 x5 = x3 x0 x5 ⟶ x2 x4 x5 = x3 x4 x5 ⟶ add_SNo (x2 x4 x1) (add_SNo (x2 x0 x5) (minus_SNo (x2 x4 x5))) = add_SNo (x3 x4 x1) (add_SNo (x3 x0 x5) (minus_SNo (x3 x4 x5))) (proof)
Theorem 712f0..^{Conj_mul_SNo_eq__19__1} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . (∀ x6 . x6 ∈ SNoS_ (SNoLev x0) ⟶ ∀ x7 . SNo x7 ⟶ x2 x6 x7 = x3 x6 x7) ⟶ SNo x5 ⟶ x2 x4 x1 = x3 x4 x1 ⟶ x2 x0 x5 = x3 x0 x5 ⟶ x2 x4 x5 = x3 x4 x5 ⟶ add_SNo (x2 x4 x1) (add_SNo (x2 x0 x5) (minus_SNo (x2 x4 x5))) = add_SNo (x3 x4 x1) (add_SNo (x3 x0 x5) (minus_SNo (x3 x4 x5))) (proof)
Theorem 712f0..^{Conj_mul_SNo_eq__19__1} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . (∀ x6 . x6 ∈ SNoS_ (SNoLev x0) ⟶ ∀ x7 . SNo x7 ⟶ x2 x6 x7 = x3 x6 x7) ⟶ SNo x5 ⟶ x2 x4 x1 = x3 x4 x1 ⟶ x2 x0 x5 = x3 x0 x5 ⟶ x2 x4 x5 = x3 x4 x5 ⟶ add_SNo (x2 x4 x1) (add_SNo (x2 x0 x5) (minus_SNo (x2 x4 x5))) = add_SNo (x3 x4 x1) (add_SNo (x3 x0 x5) (minus_SNo (x3 x4 x5))) (proof)
Theorem 68a43..^{Conj_mul_SNo_eq__19__2} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . (∀ x6 . x6 ∈ SNoS_ (SNoLev x0) ⟶ ∀ x7 . SNo x7 ⟶ x2 x6 x7 = x3 x6 x7) ⟶ x4 ∈ SNoS_ (SNoLev x0) ⟶ x2 x4 x1 = x3 x4 x1 ⟶ x2 x0 x5 = x3 x0 x5 ⟶ x2 x4 x5 = x3 x4 x5 ⟶ add_SNo (x2 x4 x1) (add_SNo (x2 x0 x5) (minus_SNo (x2 x4 x5))) = add_SNo (x3 x4 x1) (add_SNo (x3 x0 x5) (minus_SNo (x3 x4 x5))) (proof)
Theorem 68a43..^{Conj_mul_SNo_eq__19__2} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . (∀ x6 . x6 ∈ SNoS_ (SNoLev x0) ⟶ ∀ x7 . SNo x7 ⟶ x2 x6 x7 = x3 x6 x7) ⟶ x4 ∈ SNoS_ (SNoLev x0) ⟶ x2 x4 x1 = x3 x4 x1 ⟶ x2 x0 x5 = x3 x0 x5 ⟶ x2 x4 x5 = x3 x4 x5 ⟶ add_SNo (x2 x4 x1) (add_SNo (x2 x0 x5) (minus_SNo (x2 x4 x5))) = add_SNo (x3 x4 x1) (add_SNo (x3 x0 x5) (minus_SNo (x3 x4 x5))) (proof)
Theorem 68a43..^{Conj_mul_SNo_eq__19__2} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . (∀ x6 . x6 ∈ SNoS_ (SNoLev x0) ⟶ ∀ x7 . SNo x7 ⟶ x2 x6 x7 = x3 x6 x7) ⟶ x4 ∈ SNoS_ (SNoLev x0) ⟶ x2 x4 x1 = x3 x4 x1 ⟶ x2 x0 x5 = x3 x0 x5 ⟶ x2 x4 x5 = x3 x4 x5 ⟶ add_SNo (x2 x4 x1) (add_SNo (x2 x0 x5) (minus_SNo (x2 x4 x5))) = add_SNo (x3 x4 x1) (add_SNo (x3 x0 x5) (minus_SNo (x3 x4 x5))) (proof)
Theorem 68a43..^{Conj_mul_SNo_eq__19__2} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . (∀ x6 . x6 ∈ SNoS_ (SNoLev x0) ⟶ ∀ x7 . SNo x7 ⟶ x2 x6 x7 = x3 x6 x7) ⟶ x4 ∈ SNoS_ (SNoLev x0) ⟶ x2 x4 x1 = x3 x4 x1 ⟶ x2 x0 x5 = x3 x0 x5 ⟶ x2 x4 x5 = x3 x4 x5 ⟶ add_SNo (x2 x4 x1) (add_SNo (x2 x0 x5) (minus_SNo (x2 x4 x5))) = add_SNo (x3 x4 x1) (add_SNo (x3 x0 x5) (minus_SNo (x3 x4 x5))) (proof)
Theorem 59fb3..^{Conj_mul_SNo_oneR__3__0} : ∀ x0 x1 . (∀ x2 . x2 ∈ SNoL x0 ⟶ ∀ x3 . x3 ∈ SNoL 1 ⟶ SNoLt (add_SNo (mul_SNo x2 1) (mul_SNo x0 x3)) (add_SNo (mul_SNo x0 1) (mul_SNo x2 x3))) ⟶ 0 ∈ SNoL 1 ⟶ x1 ∈ SNoL x0 ⟶ SNo x1 ⟶ add_SNo (mul_SNo x1 1) (mul_SNo x0 0) = x1 ⟶ add_SNo (mul_SNo x0 1) (mul_SNo x1 0) = mul_SNo x0 1 ⟶ SNoLt x1 (mul_SNo x0 1) (proof)
Theorem e3ee4..^{Conj_SNoS_ordsucc_omega_bdd_drat_intvl__5__2} : ∀ x0 . x0 ∈ SNoS_ (ordsucc omega) ⟶ SNoLt (minus_SNo omega) x0 ⟶ SNo x0 ⟶ not (∀ x1 . x1 ∈ omega ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ SNoS_ omega) (and (SNoLt x3 x0) (SNoLt x0 (add_SNo x3 (eps_ x1)))) ⟶ x2) ⟶ x2) ⟶ nIn x0 (SNoS_ omega) ⟶ not (∀ x1 . nat_p x1 ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ SNoS_ omega) (and (SNoLt x3 x0) (SNoLt x0 (add_SNo x3 (eps_ x1)))) ⟶ x2) ⟶ x2) (proof)
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Known add_SNo_ordinal_SL^{add_SNo_ordinal_SL} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . ordinal x1 ⟶ add_SNo (ordsucc x0) x1 = ordsucc (add_SNo x0 x1)
Theorem 8c732..^{Conj_add_SNo_ordinal_SL__11__9} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ (∀ x2 . x2 ∈ x1 ⟶ add_SNo (ordsucc x0) x2 = ordsucc (add_SNo x0 x2)) ⟶ SNo x0 ⟶ SNo x1 ⟶ ordinal (add_SNo x0 x1) ⟶ ordinal (ordsucc x0) ⟶ SNo (ordsucc x0) ⟶ ordinal (add_SNo (ordsucc x0) x1) ⟶ ordsucc (add_SNo x0 x1) ∈ add_SNo (ordsucc x0) x1 ⟶ not (ordinal (ordsucc (add_SNo x0 x1))) (proof)
Known minus_SNoCut_eq^{minus_SNoCut_eq} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ minus_SNo (SNoCut x0 x1) = SNoCut (prim5 x1 minus_SNo) (prim5 x0 minus_SNo)
Theorem 9d1d7..^{Conj_minus_SNoCut_eq_lem__11__3} : ∀ x0 x1 x2 . SNo x0 ⟶ (∀ x3 . x3 ∈ SNoS_ (SNoLev x0) ⟶ ∀ x4 x5 . SNoCutP x4 x5 ⟶ x3 = SNoCut x4 x5 ⟶ minus_SNo x3 = SNoCut (prim5 x5 minus_SNo) (prim5 x4 minus_SNo)) ⟶ SNoCutP x1 x2 ⟶ (∀ x3 . x3 ∈ x2 ⟶ SNo x3) ⟶ x0 = SNoCut x1 x2 ⟶ SNo (SNoCut x1 x2) ⟶ minus_SNo x0 = SNoCut (prim5 x2 minus_SNo) (prim5 x1 minus_SNo) (proof)
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 and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
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
Known minus_SNo_Lt_contra^{minus_SNo_Lt_contra} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 x1 ⟶ SNoLt (minus_SNo x1) (minus_SNo x0)
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
Theorem e0a4a..^{Conj_minus_SNo_prop1__2__2} : ∀ x0 x1 . SNo x0 ⟶ (∀ x2 . x2 ∈ SNoS_ (SNoLev x0) ⟶ and (and (and (SNo (minus_SNo x2)) (∀ x3 . x3 ∈ SNoL x2 ⟶ SNoLt (minus_SNo x2) (minus_SNo x3))) (∀ x3 . x3 ∈ SNoR x2 ⟶ SNoLt (minus_SNo x3) (minus_SNo x2))) (SNoCutP (prim5 (SNoR x2) minus_SNo) (prim5 (SNoL x2) minus_SNo))) ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ x1 ∈ SNoS_ (SNoLev x0) ⟶ and (and (SNo (minus_SNo x1)) (∀ x2 . x2 ∈ SNoL x1 ⟶ SNoLt (minus_SNo x1) (minus_SNo x2))) (∀ x2 . x2 ∈ SNoR x1 ⟶ SNoLt (minus_SNo x2) (minus_SNo x1)) (proof)
Param SNoEq_^SNoEq_ : ι → ι → ι → ο
Known SNoCutP_SNo_SNoCut^{SNoCutP_SNo_SNoCut} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ SNo (SNoCut x0 x1)
Theorem 59be3..^{Conj_minus_SNo_invol__5__6} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ (∀ x2 . x2 ∈ x0 ⟶ minus_SNo (minus_SNo x2) = x2) ⟶ (∀ x2 . x2 ∈ x1 ⟶ minus_SNo (minus_SNo x2) = x2) ⟶ (∀ x2 . x2 ∈ x0 ⟶ SNo x2) ⟶ (∀ x2 . x2 ∈ x1 ⟶ SNo x2) ⟶ SNo (SNoCut x0 x1) ⟶ SNo (minus_SNo (minus_SNo (SNoCut x0 x1))) ⟶ and (SNoLev (SNoCut x0 x1) ⊆ SNoLev (minus_SNo (minus_SNo (SNoCut x0 x1)))) (SNoEq_ (SNoLev (SNoCut x0 x1)) (SNoCut x0 x1) (minus_SNo (minus_SNo (SNoCut x0 x1)))) ⟶ minus_SNo (minus_SNo (SNoCut x0 x1)) = SNoCut x0 x1 (proof)
Known SNoLt_tra^SNoLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt 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)
Theorem 5101e..^{Conj_mul_SNo_SNoL_interpolate__5__9} : ∀ x0 x1 x2 x3 x4 x5 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ (∀ x6 . x6 ∈ SNoL x0 ⟶ ∀ x7 . x7 ∈ SNoL x1 ⟶ SNoLt (add_SNo (mul_SNo x6 x1) (mul_SNo x0 x7)) (add_SNo x2 (mul_SNo x6 x7))) ⟶ SNo x3 ⟶ SNoLt x2 x3 ⟶ x4 ∈ SNoL x0 ⟶ x5 ∈ SNoL x1 ⟶ SNoLe (add_SNo x3 (mul_SNo x4 x5)) (add_SNo (mul_SNo x4 x1) (mul_SNo x0 x5)) ⟶ SNo x5 ⟶ SNo (mul_SNo x4 x5) ⟶ SNoLt (add_SNo x3 (mul_SNo x4 x5)) (add_SNo x2 (mul_SNo x4 x5)) ⟶ SNoLt (mul_SNo x0 x1) x3 (proof)
Theorem d0d3d..^{Conj_minus_SNo_prop1__5__9} : ∀ x0 x1 x2 . SNo x0 ⟶ (∀ x3 . x3 ∈ SNoS_ (SNoLev x0) ⟶ and (and (and (SNo (minus_SNo x3)) (∀ x4 . x4 ∈ SNoL x3 ⟶ SNoLt (minus_SNo x3) (minus_SNo x4))) (∀ x4 . x4 ∈ SNoR x3 ⟶ SNoLt (minus_SNo x4) (minus_SNo x3))) (SNoCutP (prim5 (SNoR x3) minus_SNo) (prim5 (SNoL x3) minus_SNo))) ⟶ SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x0 x1 ⟶ SNo x2 ⟶ SNoLt x2 x0 ⟶ SNo (minus_SNo x2) ⟶ (∀ x3 . x3 ∈ SNoR x2 ⟶ SNoLt (minus_SNo x3) (minus_SNo x2)) ⟶ (∀ x3 . x3 ∈ SNoL x1 ⟶ SNoLt (minus_SNo x1) (minus_SNo x3)) ⟶ SNoLt x2 x1 ⟶ SNoLt (minus_SNo x1) (minus_SNo x2) (proof)
Theorem ec8fd..^{Conj_minus_SNo_prop1__5__7} : ∀ x0 x1 x2 . SNo x0 ⟶ (∀ x3 . x3 ∈ SNoS_ (SNoLev x0) ⟶ and (and (and (SNo (minus_SNo x3)) (∀ x4 . x4 ∈ SNoL x3 ⟶ SNoLt (minus_SNo x3) (minus_SNo x4))) (∀ x4 . x4 ∈ SNoR x3 ⟶ SNoLt (minus_SNo x4) (minus_SNo x3))) (SNoCutP (prim5 (SNoR x3) minus_SNo) (prim5 (SNoL x3) minus_SNo))) ⟶ SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x0 x1 ⟶ SNo x2 ⟶ SNoLt x2 x0 ⟶ (∀ x3 . x3 ∈ SNoR x2 ⟶ SNoLt (minus_SNo x3) (minus_SNo x2)) ⟶ SNo (minus_SNo x1) ⟶ (∀ x3 . x3 ∈ SNoL x1 ⟶ SNoLt (minus_SNo x1) (minus_SNo x3)) ⟶ SNoLt x2 x1 ⟶ SNoLt (minus_SNo x1) (minus_SNo x2) (proof)
Param diadic_rational_p^{diadic_rational_p} : ι → ο
Param SNo_min_of^SNo_min_of : ι → ι → ο
Known SNoS_omega_diadic_rational_p_lem^{SNoS_omega_diadic_rational_p_lem} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 = x0 ⟶ diadic_rational_p x1
Theorem ff211..^{Conj_SNoS_omega_diadic_rational_p_lem__11__5} : ∀ x0 x1 x2 x3 . nat_p x0 ⟶ (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . SNo x5 ⟶ SNoLev x5 = x4 ⟶ diadic_rational_p x5) ⟶ SNo x1 ⟶ SNoLev x1 = x0 ⟶ not (diadic_rational_p x1) ⟶ SNo x2 ⟶ SNoLev x2 ∈ SNoLev x1 ⟶ SNoLt x2 x1 ⟶ SNo_min_of (SNoR x1) x3 ⟶ SNo x3 ⟶ SNoLev x3 ∈ SNoLev x1 ⟶ SNoLt x1 x3 ⟶ SNo (add_SNo x1 x1) ⟶ SNo (add_SNo x2 x3) ⟶ not (diadic_rational_p x2) (proof)
Param abs_SNo^abs_SNo : ι → ι
Param ap^ap : ι → ι → ι
Theorem 694cf..^{Conj_SNo_approx_real__10__10} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo (minus_SNo x0) ⟶ x2 ∈ SNoS_ omega ⟶ (∀ x4 . x4 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x2 (minus_SNo x0))) (eps_ x4)) ⟶ SNo x2 ⟶ SNo (add_SNo x2 (minus_SNo x0)) ⟶ (∀ x4 . x4 ∈ SNoS_ omega ⟶ (∀ x5 . x5 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x4 (minus_SNo (ap x1 (ordsucc x3))))) (eps_ x5)) ⟶ x4 = ap x1 (ordsucc x3)) ⟶ SNo (ap x1 (ordsucc x3)) ⟶ SNoLt (ap x1 (ordsucc x3)) x2 ⟶ SNoLt 0 (add_SNo x2 (minus_SNo (ap x1 (ordsucc x3)))) ⟶ SNoLt x0 (ap x1 (ordsucc x3)) ⟶ abs_SNo (add_SNo x2 (minus_SNo x0)) = add_SNo x2 (minus_SNo x0) ⟶ x2 = ap x1 (ordsucc x3) ⟶ SNoLt x2 (ap x1 x3) (proof)
Param SNo_max_of^SNo_max_of : ι → ι → ο
Theorem 9e6ca..^{Conj_SNoS_omega_diadic_rational_p_lem__10__12} : ∀ x0 x1 x2 x3 . nat_p x0 ⟶ (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . SNo x5 ⟶ SNoLev x5 = x4 ⟶ diadic_rational_p x5) ⟶ SNo x1 ⟶ SNoLev x1 = x0 ⟶ not (diadic_rational_p x1) ⟶ SNo_max_of (SNoL x1) x2 ⟶ SNo x2 ⟶ SNoLev x2 ∈ SNoLev x1 ⟶ SNoLt x2 x1 ⟶ SNo_min_of (SNoR x1) x3 ⟶ SNo x3 ⟶ SNoLev x3 ∈ SNoLev x1 ⟶ SNo (add_SNo x1 x1) ⟶ SNo (add_SNo x2 x3) ⟶ diadic_rational_p x2 ⟶ not (diadic_rational_p x3) (proof)
Theorem 2167e..^{Conj_SNoS_omega_diadic_rational_p_lem__10__10} : ∀ x0 x1 x2 x3 . nat_p x0 ⟶ (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . SNo x5 ⟶ SNoLev x5 = x4 ⟶ diadic_rational_p x5) ⟶ SNo x1 ⟶ SNoLev x1 = x0 ⟶ not (diadic_rational_p x1) ⟶ SNo_max_of (SNoL x1) x2 ⟶ SNo x2 ⟶ SNoLev x2 ∈ SNoLev x1 ⟶ SNoLt x2 x1 ⟶ SNo_min_of (SNoR x1) x3 ⟶ SNoLev x3 ∈ SNoLev x1 ⟶ SNoLt x1 x3 ⟶ SNo (add_SNo x1 x1) ⟶ SNo (add_SNo x2 x3) ⟶ diadic_rational_p x2 ⟶ not (diadic_rational_p x3) (proof)
Param lam^Sigma : ι → (ι → ι) → ι
Theorem d22e6..^{Conj_real_add_SNo__6__10} : ∀ x0 x1 x2 x3 x4 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1) ⟶ (∀ x5 . x5 ∈ SNoS_ omega ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x5 (minus_SNo x1))) (eps_ x6)) ⟶ x5 = x1) ⟶ (∀ x5 . x5 ∈ omega ⟶ SNo (ap (lam omega (λ x6 . add_SNo (ap x2 (ordsucc x6)) (ap x3 (ordsucc x6)))) x5)) ⟶ (∀ x5 . x5 ∈ omega ⟶ SNoLt (add_SNo (ap (lam omega (λ x6 . add_SNo (ap x2 (ordsucc x6)) (ap x3 (ordsucc x6)))) x5) (minus_SNo (eps_ x5))) (add_SNo x0 x1)) ⟶ SNo x4 ⟶ SNoLt x1 x4 ⟶ x4 ∈ SNoS_ omega ⟶ not (∀ x5 : ο . (∀ x6 . and (x6 ∈ omega) (SNoLe (ap (lam omega (λ x7 . add_SNo (ap x2 (ordsucc x7)) (ap x3 (ordsucc x7)))) x6) (add_SNo x0 x4)) ⟶ x5) ⟶ x5) ⟶ x4 = x1 ⟶ ∀ x5 : ο . x5 (proof)
Theorem fe17b..^{Conj_real_add_SNo__18__2} : ∀ x0 x1 x2 x3 x4 . SNo x0 ⟶ SNo x1 ⟶ (∀ x5 . x5 ∈ SNoS_ omega ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x5 (minus_SNo x0))) (eps_ x6)) ⟶ x5 = x0) ⟶ (∀ x5 . x5 ∈ omega ⟶ SNo (ap (lam omega (λ x6 . add_SNo (ap x2 (ordsucc x6)) (ap x3 (ordsucc x6)))) x5)) ⟶ (∀ x5 . x5 ∈ omega ⟶ SNoLt (add_SNo x0 x1) (add_SNo (ap (lam omega (λ x6 . add_SNo (ap x2 (ordsucc x6)) (ap x3 (ordsucc x6)))) x5) (eps_ x5))) ⟶ SNo x4 ⟶ SNoLt x4 x0 ⟶ x4 ∈ SNoS_ omega ⟶ not (∀ x5 : ο . (∀ x6 . and (x6 ∈ omega) (SNoLe (add_SNo x4 x1) (ap (lam omega (λ x7 . add_SNo (ap x2 (ordsucc x7)) (ap x3 (ordsucc x7)))) x6)) ⟶ x5) ⟶ x5) ⟶ SNoLt 0 (add_SNo x0 (minus_SNo x4)) ⟶ x4 = x0 ⟶ ∀ x5 : ο . x5 (proof)
Theorem 06412..^{Conj_real_add_SNo__6__6} : ∀ x0 x1 x2 x3 x4 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1) ⟶ (∀ x5 . x5 ∈ SNoS_ omega ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x5 (minus_SNo x1))) (eps_ x6)) ⟶ x5 = x1) ⟶ (∀ x5 . x5 ∈ omega ⟶ SNo (ap (lam omega (λ x6 . add_SNo (ap x2 (ordsucc x6)) (ap x3 (ordsucc x6)))) x5)) ⟶ (∀ x5 . x5 ∈ omega ⟶ SNoLt (add_SNo (ap (lam omega (λ x6 . add_SNo (ap x2 (ordsucc x6)) (ap x3 (ordsucc x6)))) x5) (minus_SNo (eps_ x5))) (add_SNo x0 x1)) ⟶ SNoLt x1 x4 ⟶ x4 ∈ SNoS_ omega ⟶ not (∀ x5 : ο . (∀ x6 . and (x6 ∈ omega) (SNoLe (ap (lam omega (λ x7 . add_SNo (ap x2 (ordsucc x7)) (ap x3 (ordsucc x7)))) x6) (add_SNo x0 x4)) ⟶ x5) ⟶ x5) ⟶ SNoLt 0 (add_SNo x4 (minus_SNo x1)) ⟶ x4 = x1 ⟶ ∀ x5 : ο . x5 (proof)
Theorem 35556..^{Conj_real_add_SNo__18__6} : ∀ x0 x1 x2 x3 x4 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1) ⟶ (∀ x5 . x5 ∈ SNoS_ omega ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x5 (minus_SNo x0))) (eps_ x6)) ⟶ x5 = x0) ⟶ (∀ x5 . x5 ∈ omega ⟶ SNo (ap (lam omega (λ x6 . add_SNo (ap x2 (ordsucc x6)) (ap x3 (ordsucc x6)))) x5)) ⟶ (∀ x5 . x5 ∈ omega ⟶ SNoLt (add_SNo x0 x1) (add_SNo (ap (lam omega (λ x6 . add_SNo (ap x2 (ordsucc x6)) (ap x3 (ordsucc x6)))) x5) (eps_ x5))) ⟶ SNoLt x4 x0 ⟶ x4 ∈ SNoS_ omega ⟶ not (∀ x5 : ο . (∀ x6 . and (x6 ∈ omega) (SNoLe (add_SNo x4 x1) (ap (lam omega (λ x7 . add_SNo (ap x2 (ordsucc x7)) (ap x3 (ordsucc x7)))) x6)) ⟶ x5) ⟶ x5) ⟶ SNoLt 0 (add_SNo x0 (minus_SNo x4)) ⟶ x4 = x0 ⟶ ∀ x5 : ο . x5 (proof)
Known add_SNo_minus_Lt2^{add_SNo_minus_Lt2} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x2 (add_SNo x0 (minus_SNo x1)) ⟶ SNoLt (add_SNo x2 x1) x0
Theorem 1dbb1..^{Conj_real_add_SNo__10__7} : ∀ x0 x1 x2 x3 x4 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1) ⟶ (∀ x5 . x5 ∈ SNoS_ omega ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x5 (minus_SNo x0))) (eps_ x6)) ⟶ x5 = x0) ⟶ (∀ x5 . x5 ∈ omega ⟶ SNo (ap (lam omega (λ x6 . add_SNo (ap x2 (ordsucc x6)) (ap x3 (ordsucc x6)))) x5)) ⟶ (∀ x5 . x5 ∈ omega ⟶ SNoLt (add_SNo (ap (lam omega (λ x6 . add_SNo (ap x2 (ordsucc x6)) (ap x3 (ordsucc x6)))) x5) (minus_SNo (eps_ x5))) (add_SNo x0 x1)) ⟶ SNo x4 ⟶ x4 ∈ SNoS_ omega ⟶ not (∀ x5 : ο . (∀ x6 . and (x6 ∈ omega) (SNoLe (ap (lam omega (λ x7 . add_SNo (ap x2 (ordsucc x7)) (ap x3 (ordsucc x7)))) x6) (add_SNo x4 x1)) ⟶ x5) ⟶ x5) ⟶ SNoLt 0 (add_SNo x4 (minus_SNo x0)) ⟶ x4 = x0 ⟶ ∀ x5 : ο . x5 (proof)
Known SNo_abs_SNo^SNo_abs_SNo : ∀ x0 . SNo x0 ⟶ SNo (abs_SNo x0)
Theorem 09057..^{Conj_real_mul_SNo_pos__14__9} : ∀ x0 x1 x2 x3 x4 x5 x6 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1) ⟶ SNo (minus_SNo (mul_SNo x0 x1)) ⟶ SNo x2 ⟶ SNoLt (mul_SNo x0 x1) x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ SNoLe (add_SNo (mul_SNo x3 x1) (mul_SNo x0 x4)) (add_SNo x2 (mul_SNo x3 x4)) ⟶ SNo (mul_SNo x0 x4) ⟶ SNo (mul_SNo x3 x4) ⟶ SNo (minus_SNo (mul_SNo x3 x4)) ⟶ SNo (add_SNo x0 (minus_SNo x3)) ⟶ SNo (add_SNo x4 (minus_SNo x1)) ⟶ x5 ∈ omega ⟶ SNoLe (eps_ x5) (add_SNo x0 (minus_SNo x3)) ⟶ x6 ∈ omega ⟶ SNoLe (eps_ x6) (add_SNo x4 (minus_SNo x1)) ⟶ SNo (eps_ x5) ⟶ SNo (eps_ x6) ⟶ SNo (eps_ (add_SNo x5 x6)) ⟶ SNo (mul_SNo (eps_ x5) (eps_ x6)) ⟶ SNoLt (abs_SNo (add_SNo x2 (minus_SNo (mul_SNo x0 x1)))) (eps_ (add_SNo x5 x6)) ⟶ not (SNoLe (eps_ (add_SNo x5 x6)) (abs_SNo (add_SNo x2 (minus_SNo (mul_SNo x0 x1))))) (proof)
Known real_mul_SNo^real_mul_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ mul_SNo x0 x1 ∈ real
Known real_I^real_I : ∀ x0 . x0 ∈ SNoS_ (ordsucc omega) ⟶ (x0 = omega ⟶ ∀ x1 : ο . x1) ⟶ (x0 = minus_SNo omega ⟶ ∀ x1 : ο . x1) ⟶ (∀ x1 . x1 ∈ SNoS_ omega ⟶ (∀ x2 . x2 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x1 (minus_SNo x0))) (eps_ x2)) ⟶ x1 = x0) ⟶ x0 ∈ real
Known SNo_omega^SNo_omega : SNo omega
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 minus_SNo_0^minus_SNo_0 : minus_SNo 0 = 0
Known ordinal_In_SNoLt^{ordinal_In_SNoLt} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ SNoLt x1 x0
Known omega_ordinal^{omega_ordinal} : ordinal omega
Known nat_0^nat_0 : nat_p 0
Known SNoS_I^SNoS_I : ∀ x0 . ordinal x0 ⟶ ∀ x1 x2 . x2 ∈ x0 ⟶ SNo_ x2 x1 ⟶ x1 ∈ SNoS_ x0
Known ordsucc_omega_ordinal^{ordsucc_omega_ordinal} : ordinal (ordsucc omega)
Known SNoLev_^SNoLev_ : ∀ x0 . SNo x0 ⟶ SNo_ (SNoLev x0) x0
Theorem e4ae8..^{Conj_real_mul_SNo_pos__135__10} : ∀ x0 x1 . SNoLt 0 x0 ⟶ SNoLt 0 x1 ⟶ not (mul_SNo x0 x1 ∈ real) ⟶ SNo x0 ⟶ SNoLev x0 ∈ ordsucc omega ⟶ x0 ∈ SNoS_ (ordsucc omega) ⟶ SNoLt x0 omega ⟶ (∀ x2 . x2 ∈ SNoS_ omega ⟶ (∀ x3 . x3 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x2 (minus_SNo x0))) (eps_ x3)) ⟶ x2 = x0) ⟶ SNo x1 ⟶ SNoLev x1 ∈ ordsucc omega ⟶ SNoLt x1 omega ⟶ (∀ x2 . x2 ∈ SNoS_ omega ⟶ (∀ x3 . x3 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x2 (minus_SNo x1))) (eps_ x3)) ⟶ x2 = x1) ⟶ (∀ x2 . x2 ∈ omega ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ SNoS_ omega ⟶ SNoLt 0 x4 ⟶ SNoLt x4 x0 ⟶ SNoLt x0 (add_SNo x4 (eps_ x2)) ⟶ x3) ⟶ x3) ⟶ (∀ x2 . x2 ∈ omega ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ SNoS_ omega ⟶ SNoLt 0 x4 ⟶ SNoLt x4 x1 ⟶ SNoLt x1 (add_SNo x4 (eps_ x2)) ⟶ x3) ⟶ x3) ⟶ not (SNo (mul_SNo x0 x1)) (proof)
Theorem da648..^{Conj_minus_SNoCut_eq_lem__8__3} : ∀ x0 x1 x2 . SNo x0 ⟶ (∀ x3 . x3 ∈ SNoS_ (SNoLev x0) ⟶ ∀ x4 x5 . SNoCutP x4 x5 ⟶ x3 = SNoCut x4 x5 ⟶ minus_SNo x3 = SNoCut (prim5 x5 minus_SNo) (prim5 x4 minus_SNo)) ⟶ SNoCutP x1 x2 ⟶ (∀ x3 . x3 ∈ x2 ⟶ SNo x3) ⟶ x0 = SNoCut x1 x2 ⟶ SNoCutP (prim5 x2 minus_SNo) (prim5 x1 minus_SNo) ⟶ SNo (SNoCut (prim5 x2 minus_SNo) (prim5 x1 minus_SNo)) ⟶ (∀ x3 . SNo x3 ⟶ (∀ x4 . x4 ∈ prim5 x2 minus_SNo ⟶ SNoLt x4 x3) ⟶ (∀ x4 . x4 ∈ prim5 x1 minus_SNo ⟶ SNoLt x3 x4) ⟶ and (SNoLev (SNoCut (prim5 x2 minus_SNo) (prim5 x1 minus_SNo)) ⊆ SNoLev x3) (SNoEq_ (SNoLev (SNoCut (prim5 x2 minus_SNo) (prim5 x1 minus_SNo))) (SNoCut (prim5 x2 minus_SNo) (prim5 x1 minus_SNo)) x3)) ⟶ (∀ x3 . x3 ∈ prim5 x2 minus_SNo ⟶ SNoLt x3 (minus_SNo x0)) ⟶ (∀ x3 . x3 ∈ prim5 x1 minus_SNo ⟶ SNoLt (minus_SNo x0) x3) ⟶ minus_SNo x0 = SNoCut (prim5 x2 minus_SNo) (prim5 x1 minus_SNo) (proof)
Theorem 9e3c5..^{Conj_real_mul_SNo_pos__132__4} : ∀ x0 x1 . SNoLt 0 x0 ⟶ SNoLt 0 x1 ⟶ not (mul_SNo x0 x1 ∈ real) ⟶ SNo x0 ⟶ x0 ∈ SNoS_ (ordsucc omega) ⟶ SNoLt x0 omega ⟶ (∀ x2 . x2 ∈ SNoS_ omega ⟶ (∀ x3 . x3 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x2 (minus_SNo x0))) (eps_ x3)) ⟶ x2 = x0) ⟶ SNo x1 ⟶ SNoLev x1 ∈ ordsucc omega ⟶ x1 ∈ SNoS_ (ordsucc omega) ⟶ SNoLt x1 omega ⟶ (∀ x2 . x2 ∈ SNoS_ omega ⟶ (∀ x3 . x3 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x2 (minus_SNo x1))) (eps_ x3)) ⟶ x2 = x1) ⟶ (∀ x2 . x2 ∈ omega ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ SNoS_ omega ⟶ SNoLt 0 x4 ⟶ SNoLt x4 x0 ⟶ SNoLt x0 (add_SNo x4 (eps_ x2)) ⟶ x3) ⟶ x3) ⟶ (∀ x2 . x2 ∈ omega ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ SNoS_ omega ⟶ SNoLt 0 x4 ⟶ SNoLt x4 x1 ⟶ SNoLt x1 (add_SNo x4 (eps_ x2)) ⟶ x3) ⟶ x3) ⟶ SNo (mul_SNo x0 x1) ⟶ SNo (minus_SNo (mul_SNo x0 x1)) ⟶ nIn (SNoLev (mul_SNo x0 x1)) omega ⟶ not (∀ x2 . SNo x2 ⟶ SNoLev x2 ∈ omega ⟶ SNoLev x2 ∈ SNoLev (mul_SNo x0 x1)) (proof)
Param setexp^setexp : ι → ι → ι
Known real_add_SNo^real_add_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ add_SNo x0 x1 ∈ real
Theorem aa8d2..^{Conj_real_add_SNo__45__16} : ∀ x0 x1 x2 x3 x4 x5 . x0 ∈ real ⟶ x1 ∈ real ⟶ x2 ∈ setexp (SNoS_ omega) omega ⟶ x3 ∈ setexp (SNoS_ omega) omega ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x2 x6) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (add_SNo (ap x2 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x2 x7) (ap x2 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x3 x6) (minus_SNo (eps_ x6))) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (ap x3 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x3 x6) (ap x3 x7)) ⟶ x4 ∈ setexp (SNoS_ omega) omega ⟶ x5 ∈ setexp (SNoS_ omega) omega ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x4 x6) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (add_SNo (ap x4 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x4 x7) (ap x4 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x5 x6) (minus_SNo (eps_ x6))) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x5 x6) (ap x5 x7)) ⟶ SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x0))) (eps_ x7)) ⟶ x6 = x0) ⟶ add_SNo x0 x1 ∈ real (proof)
Theorem 5103b..^{Conj_real_add_SNo__45__20} : ∀ x0 x1 x2 x3 x4 x5 . x0 ∈ real ⟶ x1 ∈ real ⟶ x2 ∈ setexp (SNoS_ omega) omega ⟶ x3 ∈ setexp (SNoS_ omega) omega ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x2 x6) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (add_SNo (ap x2 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x2 x7) (ap x2 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x3 x6) (minus_SNo (eps_ x6))) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (ap x3 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x3 x6) (ap x3 x7)) ⟶ x4 ∈ setexp (SNoS_ omega) omega ⟶ x5 ∈ setexp (SNoS_ omega) omega ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x4 x6) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (add_SNo (ap x4 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x4 x7) (ap x4 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x5 x6) (minus_SNo (eps_ x6))) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (ap x5 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x5 x6) (ap x5 x7)) ⟶ SNo x0 ⟶ SNo x1 ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x0))) (eps_ x7)) ⟶ x6 = x0) ⟶ add_SNo x0 x1 ∈ real (proof)
Theorem 7b761..^{Conj_mul_SNo_com__1__0} : ∀ x0 x1 x2 x3 x4 x5 . SNo x1 ⟶ (∀ x6 . x6 ∈ SNoS_ (SNoLev x0) ⟶ mul_SNo x6 x1 = mul_SNo x1 x6) ⟶ (∀ x6 . x6 ∈ SNoS_ (SNoLev x1) ⟶ mul_SNo x0 x6 = mul_SNo x6 x0) ⟶ (∀ x6 . x6 ∈ SNoS_ (SNoLev x0) ⟶ ∀ x7 . x7 ∈ SNoS_ (SNoLev x1) ⟶ mul_SNo x6 x7 = mul_SNo x7 x6) ⟶ (∀ x6 . x6 ∈ x3 ⟶ ∀ x7 : ο . (∀ x8 . x8 ∈ SNoL x0 ⟶ ∀ x9 . x9 ∈ SNoR x1 ⟶ x6 = add_SNo (mul_SNo x8 x1) (add_SNo (mul_SNo x0 x9) (minus_SNo (mul_SNo x8 x9))) ⟶ x7) ⟶ (∀ x8 . x8 ∈ SNoR x0 ⟶ ∀ x9 . x9 ∈ SNoL x1 ⟶ x6 = add_SNo (mul_SNo x8 x1) (add_SNo (mul_SNo x0 x9) (minus_SNo (mul_SNo x8 x9))) ⟶ x7) ⟶ x7) ⟶ (∀ x6 . x6 ∈ SNoL x0 ⟶ ∀ x7 . x7 ∈ SNoR x1 ⟶ add_SNo (mul_SNo x6 x1) (add_SNo (mul_SNo x0 x7) (minus_SNo (mul_SNo x6 x7))) ∈ x3) ⟶ (∀ x6 . x6 ∈ SNoR x0 ⟶ ∀ x7 . x7 ∈ SNoL x1 ⟶ add_SNo (mul_SNo x6 x1) (add_SNo (mul_SNo x0 x7) (minus_SNo (mul_SNo x6 x7))) ∈ x3) ⟶ (∀ x6 . x6 ∈ x5 ⟶ ∀ x7 : ο . (∀ x8 . x8 ∈ SNoL x1 ⟶ ∀ x9 . x9 ∈ SNoR x0 ⟶ x6 = add_SNo (mul_SNo x8 x0) (add_SNo (mul_SNo x1 x9) (minus_SNo (mul_SNo x8 x9))) ⟶ x7) ⟶ (∀ x8 . x8 ∈ SNoR x1 ⟶ ∀ x9 . x9 ∈ SNoL x0 ⟶ x6 = add_SNo (mul_SNo x8 x0) (add_SNo (mul_SNo x1 x9) (minus_SNo (mul_SNo x8 x9))) ⟶ x7) ⟶ x7) ⟶ (∀ x6 . x6 ∈ SNoL x1 ⟶ ∀ x7 . x7 ∈ SNoR x0 ⟶ add_SNo (mul_SNo x6 x0) (add_SNo (mul_SNo x1 x7) (minus_SNo (mul_SNo x6 x7))) ∈ x5) ⟶ (∀ x6 . x6 ∈ SNoR x1 ⟶ ∀ x7 . x7 ∈ SNoL x0 ⟶ add_SNo (mul_SNo x6 x0) (add_SNo (mul_SNo x1 x7) (minus_SNo (mul_SNo x6 x7))) ∈ x5) ⟶ x2 = x4 ⟶ x3 = x5 ⟶ SNoCut x2 x3 = SNoCut x4 x5 (proof)
Theorem 7312b..^{Conj_real_add_SNo__44__17} : ∀ x0 x1 x2 x3 x4 x5 . x0 ∈ real ⟶ x1 ∈ real ⟶ x2 ∈ setexp (SNoS_ omega) omega ⟶ x3 ∈ setexp (SNoS_ omega) omega ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x2 x6) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (add_SNo (ap x2 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x2 x7) (ap x2 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x3 x6) (minus_SNo (eps_ x6))) x0) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x0 (ap x3 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x3 x6) (ap x3 x7)) ⟶ x4 ∈ setexp (SNoS_ omega) omega ⟶ x5 ∈ setexp (SNoS_ omega) omega ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (ap x4 x6) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (add_SNo (ap x4 x6) (eps_ x6))) ⟶ (∀ x6 . x6 ∈ omega ⟶ ∀ x7 . x7 ∈ x6 ⟶ SNoLt (ap x4 x7) (ap x4 x6)) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt (add_SNo (ap x5 x6) (minus_SNo (eps_ x6))) x1) ⟶ (∀ x6 . x6 ∈ omega ⟶ SNoLt x1 (ap x5 x6)) ⟶ SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x0))) (eps_ x7)) ⟶ x6 = x0) ⟶ (∀ x6 . x6 ∈ SNoS_ omega ⟶ (∀ x7 . x7 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x6 (minus_SNo x1))) (eps_ x7)) ⟶ x6 = x1) ⟶ add_SNo x0 x1 ∈ real (proof)